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Algebraic Functions

- 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
- 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 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
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- 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
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- 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
- 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
- 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
- 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 Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- 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
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

You can have a mathematical problem involving both known and unknown values. For example, if you know that the age of your uncle John is twice your age plus four years, and you know that your age is 15, then you can use an algebraic function to work out the age of your uncle.

In this article, we will define what algebraic functions are, the different types of algebraic functions, how to identify algebraic and non-algebraic functions, touch on the differential calculus of algebraic functions, and work through some examples.

- What are algebraic functions?
- Types of algebraic functions
- Algebraic functions vs. non-algebraic functions
- Differential calculus with algebraic functions
- Graphing algebraic functions
- Algebraic functions example problems

As we learned from our Functions article, there are many classes of functions. One of those classes is **algebraic functions**.

An **Algebraic Function** is a function that involves only the algebraic operations: addition, subtraction, multiplication, division, powers, and roots.

We created this class of functions when we allowed for **quotients **and **fractional powers** in **polynomial functions**. If it weren't for these allowances, we would simply have polynomial functions! These additions to the polynomial functions give rise to the **types of algebraic functions**:

- Polynomial functions
- Rational functions
- Power/Root functions
- This includes functions with fractional powers because they can be written as roots. For example:

Based on our definition of an algebraic function, let's list some **examples of algebraic functions**.

Again, note that **algebraic functions include only the operations**: , **integer exponents** and **rational exponents**.

The image below shows the types of algebraic functions.

Algebraic functions can take on the form of **P****olynomial Functions**:

Where,

, , … , are all real number constants

is a positive integer

Some examples of **polynomial algebraic functions** are:

A linear function, like:

A quadratic function, like:

A cubic function, like:

A biquadratic/quartic function, like:

A quintic function, like:

Algebraic functions can take on the form of **Rational Functions**:

Where,

and are polynomials.

.

Some examples of **rational algebraic functions** are:

Algebraic functions can take on the form of **Power Functions**:

Where,

and are any real numbers.

Some examples of **power algebraic functions** are:

The reciprocal function: (note that this is also a rational function).

Be careful here! While they look similar, an **exponential function like , is not a power function**! Power functions must have the variable as the base, while exponential functions have the variable as the exponent.

A specific type of power function is the **Root Function**:

What makes this a special kind of power function?

A

**root function**is a**power function with fractional exponents**, where,is any real number.

and are positive integers greater than 1.

**does not equal**.

Some examples of root algebraic functions are:

So, if an algebraic function can include only the operations , exponents (and roots), then what about all those other functions we've learned about? Those are all **non-algebraic functions**!

Some **examples of non-algebraic functions** are:

Trigonometric functions, like:

Hyperbolic functions, like :

Exponential functions, like :

Logarithmic functions, like :

Absolute value functions, like :

When it comes to graphing algebraic functions, sometimes we need to take **Derivatives **and **Higher-Order Derivatives** to find the **critical points **and **inflection points** of the graphs of these functions.

If we want to find the **critical points** on the graph of an algebraic function, we must know how to take the derivative of the function. This can be found in our article on Derivatives.

What is a critical point?

Let be an interior point in the domain of. We say that is a **critical point** of the function if:

- exists and if either:
- or
- doesn't exist

f(c) MUST EXIST for x=c to be a critical point!

But **what is a critical point**, exactly?

Wherever we have a critical point of a function, there is either a **horizontal tangent**, a **vertical tangent**, a **sharp turn**, or a **change in concavity** at that point.

- If the critical point is , we have a horizontal tangent at that point. This means that either:
- The function has a
**maximum**or**minimum**point at \( c \).- If we have \( f(x)=x^{2} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and an absolute minimum at \( x=0 \).

- Or the function has an
**inflection point**at \( c \).- If we have \( f(x)=x^{3} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and a change in concavity, but no minimum or maximum.

- The function has a

- If the critical point is , we have either:
- A
**vertical tangent**at point \( c \).- If we have \( f(x)=x^{\frac{1}{3}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a vertical tangent and a change in concavity.

- Or a
**sharp turn**at point \( c \).- If we have \( f(x)=\sqrt[3]{x^{2}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a sharp turn and a change in concavity.

- A

To **find a critical point** of a function, we follow these steps:

Find the Derivative of the function.

Set the derivative equal to 0.

Solve for x, if possible.

Find all values of x (if any) where the derivative is undefined.

All the values of x (

**if they are within the domain of the original function**) found in steps 2 and 3 are the x-coordinates of the critical points.Find the y-values of the critical points by plugging the x-values into the original function and solving for the y-coordinate.

If we want to find the **inflection ****points** on the graph of an algebraic function, we must know how to take the second derivative of the function. This can be found in our article on Higher-Order Derivatives.

What is an inflection point?

If is continuous at and changes concavity at , then the point is an **inflection point** of .

But **what is an inflection point**, exactly?

Wherever we have an inflection point on a function, that is where it changes from either:

- Convex to Concave, or
- Concave to Convex

And what do Concave and Convex mean?^{ 1}

**Convex**(also called**Concave Upward**or Convex Downward) is when the**slope of a function is increasing**.**Concave**(also called**Concave Downward**or Convex Upward) is when the**slope of a function is decreasing**.

Note that the AP Calculus Exams (and likely your AP Calculus teacher) use the terms **Concave Up** and **Concave Down**.

So, how do we find where a function changes concavity?

We find the second derivative of the function.

If the

**second derivative**of a function is**positive**, then it is**Convex**.If the

**second derivative**of a function is**negative**, then it is**Concave**.

The graphs of all these types of algebraic functions vary widely from each other. However, the general procedure to graph an algebraic function is as follows:

Calculate the x-intercepts by setting and solving for x.

Calculate the y-intercepts by setting and solving for y.

Find and plot any asymptotes.

Find and plot any critical points.

Find and plot any inflection points.

Calculate some extra points along the curve until you get a good feel for how the curve looks.

Plot all these points and draw the curve that connects them.

Graph the function:

**Solution**:

Calculate the x-intercepts by setting and solving for x.

→ can be replaced with.

→ set and cross-multiply to solve for.

→ There are

**no x-intercepts**.

Calculate the y-intercepts by setting and solving for y.

→ can be replaced with.

→ set and solve for.

→ you can't divide by 0! There are

**no y-intercepts**.

Find and plot any asymptotes.

Since the expression as, there

**is a horizontal asymptote at the line (the x-axis)**.Since the expression as, there

**is a vertical asymptote at the line (the y-axis)**.

Find and plot any critical points.

Find the derivative of the function (check out our article on Derivatives for more info):

Set the derivative equal to 0, and solve for x.

→ while the point where is a possible candidate for a critical point because , there are

**no critical points**because we proved earlier that does not exist.

Find and plot any inflection points.

Find the second derivative of the function (check out our article on Higher-Order Derivatives for more info):

Set the second derivative equal to 0, and solve for x.

→ the second derivative is never 0, and it is undefined when. There are

**no inflection points**because the original function is not defined at 0.

Calculate some extra points along the curve until you get a good feel for how the curve looks.

x 1/x -4 -1/4 -3 -1/3 -2 -1/2 -1 -1 0 undefined 1 1 2 1/2 3 1/3 4 1/4

Plot all these points and draw the curve that connects them.

Finding the domain and range of algebraic functions depends on the type of algebraic function we are considering.

For polynomial functions:

The domain for all polynomial functions is all real numbers:

The range for polynomial functions depends on both the order of the polynomial and the y-values of the graph.

If the order of the polynomial is odd, then the range is always .

If the order of the polynomial is even, then the range depends on the minimum and/or maximum y-value(s).

Find the domain and range of the function:

**Solution**:

Since this is a polynomial function:

- The
**domain is all real numbers**:.

To find the range, we recognize that this is the equation of a parabola. We can rewrite the equation in vertex form as:

Where,

- is the leading coefficient
- is the vertex of the parabola

Since the leading coefficient is positive, we know the parabola opens upward. This means the vertex is the lowest point of the parabola.

Therefore, **the range is** .

We can graph the function to double-check our work:

For rational functions:

Finding the domain of rational functions requires that we follow the rule that the denominator cannot equal 0.

Finding the range of rational functions requires that we:

First, solve the function for x

Then, apply the rule that the denominator cannot equal 0.

Note that if the polynomials within the rational function have orders higher than 2, this process quickly becomes difficult. To find the range of rational functions whose orders are 3 or higher, we would need more differential analysis.

Find the domain and range of the function:

**Solution**:

Since this is a rational function, we must follow the rule that the denominator cannot equal 0. Therefore, the domain is the set of all real numbers except where the denominator equals 0.

**Domain**:

To find the range, we need to find the values of y for which there exists a real number of x such that:

- Multiply both sides of the equation by.
- Move all terms with an x to the right of the equals sign. Move all terms without an x to the left of the equals sign.
- Factor out the x on the right side of the equation.
- Solve for x.
- If, the equation has no solution because that would set the denominator equal to 0. Therefore, the range is:
**Range**:

We can graph the function to double-check our work:

For power/root functions:

Both domain and range of power and root functions depend on the exponent. We need to analyze each function on a case-by-case basis, as the domain and range of these functions are pretty “sensitive” to changes in exponents.

Find the domain and range of the function:

**Solution**:

To find the domain:

- Since this is a square root function, we need to keep what is inside the radical greater than or equal to 0.
- So, .

- Factor what is under the radical.
- → this inequality holds only if both terms are positive or both terms are negative.
- For both terms positive:
- Therefore, must be part of the domain.

- For both terms negative:
- There are no values of x that can satisfy both of these inequalities.

- For both terms positive:

- → this inequality holds only if both terms are positive or both terms are negative.
**Domain**:

To find the range:

- We use the domain to find the range:
- If the domain is , then .
- Therefore, .

**Range**:

We can graph the function to double-check our work:

Which of the following are algebraic functions?

**Solutions**:

- This is NOT an algebraic function.
- This is an algebraic function.
- This is NOT an algebraic function.
- This is NOT an algebraic function.
- This is an algebraic function.
- This is an algebraic function.
- This is an algebraic function.
- This is an algebraic function.
- This is NOT an algebraic function.
- This is NOT an algebraic function.
- This is an algebraic function.
- This is an algebraic function.

Find the domain of each of the following:

**Solutions**:

- We can't divide by 0, so the domain is the set of values of x such that .
**Domain**:

- Again, we can't divide by 0, so we need to determine the values of x where the denominator is 0.
- for all real numbers of x.
**Domain**:

- for all real numbers of x.
- Since the square root of a negative number is not a real number, we must determine the values of x that satisfy .
**Domain**:

- Here, the cube root is defined for all real numbers.
**Domain**:

- Algebraic functions include only the algebraic operations:
- Addition: +
- Subtraction: -
- Multiplication:
- Division:
- Powers (aka exponents):, where is any real number
- Roots:, where is any real number and is a positive integer greater than 1.

- The types of algebraic functions are:
- Polynomial functions
- Rational functions
- Power/Root functions

- Any function that contains trigonometric functions, hyperbolic functions, logarithms, variables in the power (or exponent), or an absolute value is not an algebraic function.
- Related reading:
- Functions
- Polynomial Functions
- Derivatives
- Higher-Order Derivatives

- https://math.stackexchange.com/questions/2364116/how-to-remember-which-function-is-concave-and-which-one-is-convex/2364123#2364123

Two examples of algebraic functions are rational functions and root functions.

The types of algebraic functions are:

- Polynomial Functions
- Rational Functions
- Power/Root Functions

Solving algebraic functions usually involves finding their domains and/or ranges and graphing them.

Learning algebraic functions are important for several reasons:

- Helps you develop problem-solving skills
- Learn how to structure and organize problems
- Learn how to approach problems that have no solution
- Learn to think abstractly
- Learn to visualize situations

More about Algebraic Functions

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