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Functional Analysis

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 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

What is 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 $f:\mathrm{ℝ}↦\mathrm{ℝ}$ 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.

The Domain and Range

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

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

and

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

Note that $\mathrm{ℝ}$ above represents the set of all real values that represents the interval $\left(-\infty ,\infty \right)$. 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).

Domain and Range of Polynomial Functions

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

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

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

3. 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$f\left(x\right)=a{\left(x-h\right)}^{2}+k$is

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

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

Domain and Range of Square Root Functions

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,

$x-1\ge 0⇒x\ge 1$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

Domain and Range of Cube Root Functions

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

Domain and Range of Exponential Functions

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.

Domain and Range of Logarithmic Functions

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

Domain and Range of Rational Functions

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 $\left\{x\in \mathrm{ℝ}|q\left(x\right)\ne 0\right\}$.

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.

$x+5=0⇒x=-5$

Thus, the domain is the set of all real numbers except x = –5, $\left\{x\in \mathrm{ℝ}|x\ne -5\right\}$. 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,

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

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,

$x+1=0⇒x=-1$

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 $\left\{y\in \mathrm{ℝ}|y\ne -1\right\}$. 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.

Domain and Range of Trigonometric Functions

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

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.

Even Function

A function, f is even when

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

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.

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

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

Odd Function

A function, f is odd when

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

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 180o 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

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

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

Can a Function be Both Even and Odd?

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

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

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

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

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

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.

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

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

for all values of x.

Intercept Points

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

$\left(x-8\right)\left(x+1\right)=0$

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

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

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.

Intersecting Points

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.

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

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.

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

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.

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

and

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

Thus, the points of intersection are

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

Other Components for Graph Sketching

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.

Functional Analysis - Key takeaways

• 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 $f\left(-x\right)=f\left(x\right)$for all x.
• A function is odd when $f\left(-x\right)=-f\left(x\right)$for all x.
• A function is periodic if$f\left(x+P\right)=f\left(x\right)$for all x.
• The x-intercept is a point where the function crosses the x-axis, f(x) = 0.
• The y-intercept is a point where the function crosses the y-axis, f(0).
• 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.

Functional analysis studies a function by investigating its behaviour and identifying relationships and hypotheses that may arise.

• Identifying the domain and range of a function

• Recognizing odd and even functions

• Finding the x and y-intercepts of a function

Real analysis studies the properties of numbers while functional analysis analyses the properties of functions?

Functional analysis is important so that we can make predictions and estimations for our given function

Final Functional Analysis Quiz

Question

Can a function be both concave and convex?

Yes- a straight line is both concave and convex.

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Question

Which of the following would be a convex function?

Both

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Question

Which of the following is a concave function?

Neither

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Question

Which equation is both concave and convex?

y=2x+7

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Question

A polygon with an internal angle greater than 180 degrees is:

Concave

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Question

A triangle will always be a:

convex polygon

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Question

If the graph of a function is always below a straight line joining two coordinates on the function, that function is:

Convex

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Question

If the graph of a function is always above a straight line joining two coordinates on the function, that function is:

concave

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Question

Which type of function is both concave and convex?

Straight line functions

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Question

Would a logarithmic graph be:

Concave

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Question

Would a regular hexagon be concave or convex?

Each interior angle of a regular hexagon is 120 degrees so it is concave.

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Question

Yes- we could have one angle that is 190 degrees, one angle 100 degrees, one angle 10 degrees and one angle 60 degrees.

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Question

What is growth?

Growth is a value of increase or decrease in relation to a previous value or an initial value.

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Question

Why is important to model growth?

Modelling growth is important because it helps us understand and analize growth, helping us to predict future values according to a current trend, and triggering decision-making when required.

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Question

What is a growth factor?

The growth factor is a factor by which a quantity increases or decreases per unit of another quantity. This factor can be a constant or a rate, depending on which type of growth we are dealing with.

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Question

If a population's size is increasing by a constant value per time interval, which type of growth is it experiencing?

Linear growth

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Question

If a population's size is increasing at a constant rate per time interval, what type of growth is it experiencing?

Geometric growth

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Question

When comparing the growth of polynomial functions, which one grows faster will be determined by the term with...

The highest order

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Question

What is a discrete model of growth?

A model of growth is considered discrete when the values in the model change at specific intervals and not continuously.

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Question

What is a continuous model of growth?

A model of growth is considered continuous when the values in the model change continuously and not at specific intervals.

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Question

What type of model is linear growth?

Discrete

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Question

What type of model is exponential growth?

Continuous

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Question

What type of model is logarithmic growth?

Continuous

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Question

If a population grows 25% each year, what is its growth factor?

1.25

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Question

What is compound interest?

Compound interest is another type of geometric growth model, where interest is paid on an investment and on any interest earned already.

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Question

What does the second derivative test tell us if  $$x=c$$ is a stationary point and $$f''(c)=0$$?

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Question

What does the second derivative test tell us if $$x=c$$ is a stationary point and $$f''(c)=0$$?

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Question

What is the absolute maximum of a function?

The Absolute maximum is the greatest output in its range.

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Question

What is the absolute minimum of a function?

The absolute minimum of a function is the least output in its range.

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Question

What is a relative maximum?

A relative maximum of a function is an output that is greater than the outputs next to it.

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Question

What is a relative minimum?

A relative minimum of a function is an output that is less than the outputs next to it.

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Question

Do all functions have an absolute maximum and an absolute minimum?

No. A function may keep increasing or decreasing so no absolute maximum or minimum is reached.

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Question

Where can you find the absolute maximum or the absolute minimum of a parabola?

At its vertex. If the parabola opens upwards, it is a minimum. If a parabola opens downwards, it is a maximum.

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Question

How can you identify relative minima and maxima in a graph?

The peaks of the graph are the relative maxima. The valleys are the relative minima.

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Question

What is a stationary point?

A point where the derivative (or the slope) of a function is equal to zero.

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Question

Find the stationary points of $$f(x)=x^{3}-27x$$?

$$c=3\ \text{and} \ c=-3$$

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Question

Find the absolute maximum of the function $$f(x)=e^{x}$$.

The function has no absolute maximum

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Question

Find the absolute minimum of the function $$f(x)=(x+1)^{2}-2$$

-2

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Question

Find the local maximum of the function $$f(x)=1/x-1/x^{2}$$.

$$\frac{1}{4}$$

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Question

Find the relative minimum of $$f(x)=x^{3}+1$$.

The function has no relative minimum.

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Question

What does Fermat's Theorem for stationary points state?

Fermat's Theorem for stationary points states that if a function $$f(x)$$ has a relative extremum at $$x=c$$, and the function is differentiable at that point, then $$f'(c)=0$$.

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Question

Find the vertical asymptote for the function $$y=\frac{2x+1}{x^{2}+5x+4}$$.

Equating the denominator to zero, we find that $$x^{2}+5x+4=0$$.

Solving this, we obtain the solutions $$x=-4, \ x=-1$$.

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Question

Does the function $$y=2x+1$$ have any asymptotes?

No

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Question

How many types of asymptotes can a rational function have?

Three

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Question

Name the three types of asymptotes.

Horizontal, Vertical and Slant asymptotes.

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Question

What is an asymptote?

An asymptote is a straight line that approaches a curve, but never touches the curve.

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Question

At most, how many types of asymptote can a single rational function have?

At most 2.

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Question

What are the asymptotes of $$f(x)=\frac{x-1}{x+1}$$?

$$x=-1$$ and $$y=1$$

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Question

Given the equation $$\displaystyle f(x)=\frac{2x^{2}-3x-1}{x}$$, what types of asymptotes are visible?

A Vertical Asymptote and an Oblique Asymptote.

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Question

What are the asymptotes of the equation $$\displaystyle y=\frac{x-2}{x-4}$$?

$$x=4, \ y=1$$

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Smart Formatting

Create the most beautiful study materials using our templates.