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Functional Analysis

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

Looking at functions graphically is always helpful when we need to analyze patterns and trends. Don't you agree? Graphing is widely used in many subjects such as finance, engineering, and psychology. When we plot curves on a Cartesian plane, let's say, we are able to visualize where the function might plateau or spike. By identifying such behaviors, we can make predictions and estimations for our given function. In this topic, we shall be introduced to a concept called Functional Analysis. Let us begin this topic with its definition.

**Functional analysis** is a branch of mathematics that studies functions by investigating the behavior of a given function and identifying relationships and hypotheses that may arise.

Throughout this topic, we shall only deal with real functions in one variable. By the end of this article, you should be familiar with the following concepts:

Identifying the domain and range of a function

Recognizing odd and even functions

Finding the x and y-intercepts of a function

Before we dive into this topic, let us first recall the definition of a function.

A **function **is an expression, also called a rule, that defines the relationship between one variable (independent input) and another variable (dependent output). This is commonly denoted by y = f(x) where x and y are related such that for every value of x, there is a unique value of y that obeys the rule f.

Below is an example of a function.

Say we have the function $f:\mathrm{\mathbb{R}}\mapsto \mathrm{\mathbb{R}}$ defined by

$f\left(x\right)=x+2$.This describes a function that takes a number, an input, and adds 2. For example, for an input x = 1, we have an output f(1) = 3. Similarly, for an input x = 4, we have an output f(4) = 6.

When plotting a function, it is essential to know the 'size' of the variables. This is known as the domain and range of a function. These two terms are explained below.

The **domain **of a function, f, is the set of all values for which the function is defined. The elements of the domain are represented as an independent variable, or input value, that is not dependent on any other quantity but rather, varies freely. It is often denoted by x.

The **range **of a function is the set of all resulting values that f takes, corresponding to the input values of a function. The elements of the range are dependent on the values within the domain and are sometimes referred to as the output value.

The general notation for the domain and range of a function is

$Domain\left(f\right)=\left\{x:x\in A,A\in \mathrm{\mathbb{R}}\right\}$

and

$Range\left(f\right)=\left\{f\right(x):x\in Domain(f\left)\right\}$

Note that $\mathrm{\mathbb{R}}$ above represents the set of all real values that represents the interval $(-\infty ,\infty )$. Here is a visual representation of a domain and range with regards to a function. Recall the example of the function f we had introduced earlier.

Graphical representation of a domain, range and function, StudySmarter Originals

This representation suggests that a function works like a machine that transforms elements of the domain, the inputs, into elements of the codomain. The actual outputs of this "transformation machine" will be the elements of the range, the outputs.

There are many types of functions to consider in the realm of mathematics. We have polynomial functions, exponential functions, trigonometric functions, etc. In the following sections, we shall summarize the general formulas used to find the associated domain and range of each type of function (usually seen in this syllabus).

We shall first note the three types of polynomials we shall often use throughout this topic.

linear functions, $f\left(x\right)=ax+b$

quadratic functions, $f\left(x\right)=a{x}^{2}+bx+c$

cubic functions, $f\left(x\right)=a{x}^{3}+b{x}^{2}+cx+d$

The domain of any polynomial function is the set of all real numbers, IR.

The range of a linear and cubic function is also the set of all real numbers, IR.

The range of a quadratic function of the form$f\left(x\right)=a{(x-h)}^{2}+k$is

$Range\left(f\right)=\left\{f\right(x)\ge k,ifa>0\}$ or $Range\left(f\right)=\left\{f\right(x)\le k,ifa<0\}$.

Find the domain of the function $f\left(x\right)=3x-1$.

**Solution**

This is a linear function. Thus, the domain and range of this function is the set of all real numbers, IR. The graph is shown below.

Example 1, StudySmarter Originals

For the standard square root function, $f\left(x\right)=\sqrt{x}$, the domain is the set of all real numbers, IR and the range is f(x) ≥ 0.

For a general square root function of the form $f\left(x\right)=\sqrt{g\left(x\right)}$, where g(x) is a function of x, the domain is the set of functions where g(x) ≥ 0 and the range is f(x) ≥ 0.

Determine the domain and range of the function $f\left(x\right)=\sqrt{x-1}$.

**Solution**

The domain is the set of values where the component inside the square root is more than or equal to zero, or in other words,

$x-1\ge 0\Rightarrow x\ge 1$Thus, the domain is the set of values where x is more than or equal to 1. The range is f(x) ≥ 0, for x ≥ 1. The graph is shown below.Example 2, StudySmarter Originals

For any function containing a cube root, may it be the standard form $f\left(x\right)=\sqrt[3]{x}$ or the general form $f\left(x\right)=\sqrt[3]{g\left(x\right)}$, the domain and range are both the set of all real numbers, IR.

What is the domain and range of the function $f\left(x\right)=\sqrt[3]{2-x}$.

**Solution**

The domain and range of any cube root function is the set of all real numbers, IR. Graphing this function, we find that the domain and range indeed satisfy the set of all real numbers, IR.

Example 3, StudySmarter Originals

For an exponential function of the form $f\left(x\right)={a}^{x}$, where a is any real number, the domain is the set of all real numbers, IR.

The range will always yield positive real values, that is, f(x) > 0.

Given the graph of the function $f\left(x\right)={e}^{x}$ below, determine its domain and range.

Example 4, StudySmarter Originals

**Solution**

Observing the graph above, we find that the domain satisfies the set of all real numbers. The range is f(x) > 0.

For a logarithmic function of the form $f\left(x\right)={\mathrm{log}}_{a}x$, where a is any real number, the domain is x > 0 while the range is the set of all real numbers.

The function $f\left(x\right)={\mathrm{log}}_{e}x$ can also be written as $f\left(x\right)=\mathrm{ln}\left(x\right)$. This is also known as the natural logarithm function. What is the domain and range of this function?

**Solution**

The domain here is x > 0. The range on the other hand is the set of all real numbers, IR. The graph is shown below.

Example 5, StudySmarter Originals

**Rational functions** are functions that can be represented by a rational fraction. This is generally denoted by $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, where p and q are both polynomial functions of x and q(x) **≠** 0.

The domain is the set of all real numbers except for which the denominator is equal to zero, that is $\{x\in \mathrm{\mathbb{R}}|q\left(x\right)\ne 0\}$.

The range here is the same as the domain of the inverse of this rational function f or in other words, $Range\left(f\right)=Domain\left({f}^{-1}\right)$.

Given the function $f\left(x\right)=\frac{3-x}{x+5}$, find the domain and range.

**Solution**

We shall first attempt to find the domain of this function. To find the excluded value in the domain of the function, equate the denominator to zero and solve for x.

$x+5=0\Rightarrow x=-5$

Thus, the domain is the set of all real numbers except x = –5, $\{x\in \mathrm{\mathbb{R}}|x\ne -5\}$. In other words, the graph is not defined at x = –5. Next, let us find the range by evaluating the inverse of this function. Let y = f(x). Now, interchanging the x and y from our given function, we obtain

$x=\frac{3-y}{y+5}$

Solving for y yields,

$x(y+5)=3-y\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow xy+5x=3-y\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow xy+y=3-5x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow y(x+1)=3-5x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow y=\frac{3-5x}{x+1}$

Thus, the inverse of f is

${f}^{-1}\left(x\right)=\frac{3-5x}{x+1}$

The excluded value in the domain of this inverse function can be found by equating the denominator to zero and solving for x. Thus,

$x+1=0\Rightarrow x=-1$

The domain of this inverse function is the set of real numbers except x = –1. However, in this case, this domain is the range of our function, f(x). Thus, the range of the given function is $\{y\in \mathrm{\mathbb{R}}|y\ne -1\}$. The graph is not define at y = –1. The graph is plotted below.

Example 6, StudySmarter Originals

In the graph above, the lines (in red) x = –5 and y = –1 represent the region for which the function is not defined.

Observe the graph of the sine (green line) and cosine (blue line) functions, f(x) = sin(x) and f(x) = cos(x), below.

Sine and cosine graph, StudySmarter Originals

Notice that the value of the functions oscillates between –1 and 1 and it is defined for all real numbers. Thus, for each sine and cosine function, the domain is the set of all real numbers, R and the range is –1 ≤ f(x) ≤ 1. The range here can also be denoted by [–1, 1].

Even and odd functions are functions that satisfy a particular rule of symmetry. To check whether a function is even or odd, all we need to do is substitute x into the given function and observe whether it satisfies the condition of an even or odd function, which we shall establish below. We shall look at both of these types of functions and identify their respective properties.

A function, f is **even **when

$f(-x)=f\left(x\right)$,

for all x in the domain of the function.

Geometrically speaking, the graph of an even function is symmetric with respect to the y-axis, in other words, the function remains unchanged when reflected about the y-axis. The properties of even functions include:

The sum of two even functions is even;

The difference between two even functions is even;

The product of two even functions is even;

The quotient of two even functions is even;

The derivative of an even function is odd;

The composition of two even functions is even;

The composition of an even function and odd function is even.

Let us look at an example.

Determine whether the following function is even.

$f\left(x\right)={x}^{2}+2$

**Solution **

Let us substitute –x into our function as below.

$f(-x)={(-x)}^{2}+2\Rightarrow f(-x)={x}^{2}+2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow f(-x)=f\left(x\right)$

Since $f(-x)=f\left(x\right)$, we conclude that this function is indeed an even function. The graph is shown below.

Example 7, StudySmarter Originals

Notice how the curve is reflected about the y-axis.

A function, f is **odd **when

$f(-x)=-f\left(x\right)$,

for all x in the domain of the function.

To look at this geometrically, the graph of an odd function has rotational symmetry with respect to the origin. Essentially, the function remains unchanged when rotated 180^{o} about the origin. Below are the properties of odd functions:

The sum of two odd functions is odd;

The difference between two odd functions is odd;

The product of two odd functions is even;

The product of an even function and odd function is odd;

The quotient of two odd functions is even;

The quotient of an even function and odd function is odd;

The derivative of an odd function is even;

The composition of two odd functions is odd.

Below is an example.

Verify if the following function is odd.

$f\left(x\right)=\frac{{x}^{3}-2x}{4}$

**Solution**

Plugging –x into our given function, we have

$f(-x)=\frac{{(-x)}^{3}-2(-x)}{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow f(-x)=\frac{{-x}^{3}+2x}{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow f(-x)=-\left(\frac{{x}^{3}-2x}{4}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow f(-x)=-f\left(x\right)$

Since $f(-x)=-f\left(x\right)$, we deduce that this function is odd. Below is a sketch of the graph.

Example 8, StudySmarter Originals

Notice how the curve is reflected about the origin.

There is only one function that fulfills this criterion, which is the constant function that is identically zero, f(x) = 0. The domain and range are the set of all real numbers, IR.

Note that the sum of an even function and odd function is neither even nor odd unless one of the functions is equal to zero over a given domain.

It is also possible that we have functions that are neither even nor odd. Here is an example that shows this.

Observe the following function.

$f\left(x\right)=2{x}^{2}+3x$

If we substitute –x into this function we find that we obtain a completely different function since

$f(-x)=2{(-x)}^{2}+3(-x)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow f(-x)=2{x}^{2}-3x$

Plotting the graph of f(x), observe that the graph is not reflected about the y-axis nor does it have rotational symmetry about the origin. This means that the graph is neither even nor odd.

Example 9, StudySmarter Originals

Periodic functions are used to describe trigonometric functions in particular due to the presence of oscillations and waves in their graphs.

A **periodic function** is a function that repeats itself over regular intervals (or periods). A function, f is periodic, P if

$f(x+P)=f\left(x\right)$,

for all values of x in the domain of the function. Here, $P\ne 0$ is a constant.

A function that is not periodic is termed **aperiodic**. Here is an example of that type of function.

Let us return to our sine function from the previous section.

Example 10, StudySmarter Originals

Now observe the graph above. The function repeats itself over intervals of length the sine function is periodic with period, $P=2\pi $ since

$\mathrm{sin}(x+2\pi )=\mathrm{sin}\left(x\right)$

for all values of x.

The points of interception of a function are the points at which the function crosses the axes of the graph. Below is an explicit example of the two points of interception to consider when graphing functions in two dimensions.

The **x-intercept **is a point at which the function, f crosses the x-axis. To find the x-intercept, simply solve for f(x) = 0.

The **y-intercept** is a point at which the function, f crosses the y-axis. To find the y-intercepts, substitute x = 0 into f(x).

Intercept points are important in deducing the change of sign of the curve for a given function. Let us look at an example.

Given the function below, find their x and y-intercepts.

$f\left(x\right)={x}^{2}-7x-8$

**Solution **

We begin by finding the x-intercepts. To do this, we shall equate the function to zero, f(x) = 0.

${x}^{2}-7x-8=0$

Factoring this expression, we obtain

$(x-8)(x+1)=0$

Now using the Zero Product Property and solving for x, we obtain

$x-8=0\Rightarrow x=8\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}x+1=0\Rightarrow x=-1$

Thus, we the x-intercepts are x = –1 and x = 8. Let us now look for the y-intercept. Replacing x = 0 into our function yields

$f\left(0\right)={\left(0\right)}^{2}-7\left(0\right)-8=-8$

Thus, the y-intercept is y = -8. The graph is displayed below.

Example 11, StudySmarter Originals

Notice that between the x-intercepts, x = –1 and x = 8, the function falls below the x-axis meaning that the range in this domain is negative. However, the range before x = –1 and after x = 8 are positive.

Say we are given a pair of functions, f and g. We are told to find the point(s) at which the two functions meet. This is called the intersecting point. This is defined below.

Suppose we have two functions defined by f and g. The **intersection point**(s) of these two graphs is the value(s) of x for which

$f\left(x\right)=g\left(x\right)$.

The exact value(s) of the intersection points can be found by solving the expression above algebraically. Below is an example that shows this.

Given the functions, f (in blue) and g (in red) below.

$f\left(x\right)={x}^{2}-x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}g\left(x\right)=2x+1$

Deduce their points of intersection. Both functions are plotted in the same graph below.

Example 12, StudySmarter Originals

**Solution**

Looking at the graph above, we see that there are two points of intersection for this pair of functions. We need to equate f(x) = g(x) and solve for x to find the x-coordinates for these points of intersection.

$f\left(x\right)=g\left(x\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{2}-x=2x+1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{2}-x-2-1=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{2}-3x-1=0$

Observe that we cannot factorize the equation in the final line above. In order to solve for x, we need to use the Quadratic Formula.

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{-(-3)\pm \sqrt{{(-3)}^{2}-4\left(1\right)(-1)}}{2\left(1\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{3\pm \sqrt{9+4}}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{3\pm \sqrt{13}}{2}$

Thus, we have two values of x, namely $x=\frac{3-\sqrt{13}}{2}andx=\frac{3+\sqrt{13}}{2}$. We shall leave our solution in this radical form.

To find the corresponding y-coordinates, we simply substitute these found x-values into either given function, f or g. For simplicity, we shall use the function g to find our y-values.

$g\left(\frac{3-\sqrt{13}}{2}\right)=\overline{)2}\left(\frac{3-\sqrt{13}}{\overline{)2}}\right)+1=3-\sqrt{13}+1=4-\sqrt{13}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow g\left(\frac{3-\sqrt{13}}{2}\right)=4-\sqrt{13}\phantom{\rule{0ex}{0ex}}$

and

$g\left(\frac{3+\sqrt{13}}{2}\right)=\overline{)2}\left(\frac{3+\sqrt{13}}{\overline{)2}}\right)+1=3+\sqrt{13}+1=4+\sqrt{13}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow g\left(\frac{3+\sqrt{13}}{2}\right)=4+\sqrt{13}$

Thus, the points of intersection are

$\left(\frac{3-\sqrt{13}}{2},4-\sqrt{13}\right)and\left(\frac{3+\sqrt{13}}{2},4+\sqrt{13}\right)$

So far, we have looked at the basic information needed for sketching the graph of a given function. In the following topics in this section, we shall be introduced to other fundamental elements that may be helpful when graphing functions. This includes:

Finding limits of a function

Identifying asymptotes. This is explained in the topic of Asymptotes

Locating the maximum and minimum points of a curve. This can be found here: Maxima and Minima

Using derivatives to find critical points and flex points. A detailed overview of this topic can be found here: Finding Maxima and Minima using Derivatives

By familiarizing ourselves with these methods, sketching graphs can be much more straightforward and accurate.

- The topic of functional analysis examines functions by investigating their behaviors and trends.
- The domain of a function is the set of all values for which the function is defined.
- The range of a function is the set of all resulting values that f takes, based on the domain.
- A function is even when $f(-x)=f\left(x\right)$for all x.
- A function is odd when $f(-x)=-f\left(x\right)$for all x.
- A function is periodic if$f(x+P)=f\left(x\right)$for all x.
- The x-intercept
- The y-intercept
The intersection point of two graphs is the value of x where$f\left(x\right)=g\left(x\right)$.

Plotting a function on a graph to identify its patterns and behaviours.

Identifying the domain and range of a function

Recognizing odd and even functions

Finding the x and y-intercepts of a function

More about Functional Analysis

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