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A reciprocal function can be considered as the quotient where the denominator is a polynomial with a degree of In other words, there is a variable in the denominator. In this lesson, rational functions in which the numerator and the denominator have a degree of at least will be graphed.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Average Cost Function

LaShay's school, Jeffereson High, visits a local pencil factory. The average cost, in thousand dollars, of producing thousand pencils is given by the following function.
a Graph the average cost function.
b Identify the domain and range of the function. Use a graphing calculator if necessary.
Discussion

Rational Functions

A rational function is a function that contains a rational expression. Any function that can be written as the quotient of two polynomial functions and is a rational function.

For any values of where the rational function is undefined. These values are excluded from the domain of the function. One example of a rational function is the reciprocal function.
This function has two asymptotes, the axis and the axis. The applet shows the graph of the reciprocal function and some other rational functions.
Graph of several rational functions
Discussion

Discontinuity of Rational Functions

The graph of a rational function can be a smooth continuous curve, or it can have jumps, breaks, or holes. By looking at the graph of a function, functions can be categorized into two groups: continuous or discontinuous.

Concept

Discontinuous - Function

A function is said to be continuous if its graph can be drawn without lifting the pencil. Otherwise, the function is said to be discontinuous.
Continuous and discontinuous functions
A discontinuous function can have holes, jumps, or asymptotes throughout its graph.
Examples of discontinuous functions
A point where a function is discontinuous is called a discontinuity. Function I has a point of discontinuity at Function II has a point of discontinuity at and Function III has a point of discontinuity at
Concept

Point of Discontinuity

A point of discontinuity of a function is a point with the coordinate that makes the function value undefined. A point of discontinuity can also be considered as an excluded value of the function rule.

Removable Discontinuity

If a function can be redefined so that its point of discontinuity is removed, it is called a removable discontinuity. A removable discontinuity may occur when there is a rational expression with common factors in the numerator and denominator. Consider the following rational function.
For this function, is a point of discontinuity because its denominator is when Next, notice that the function can be simplified using a difference of squares.
▼
Simplify right-hand side
WritePow

Write as a power

FacDiffSquares

SimpQuot

Simplify quotient

After simplifying, the graph behaves as a linear function except in the point of discontinuity, where its value is undefined.
Rational function with removable discontinuity
However, the function can be made continuous by redefining it at If is substituted for in the function it is found that Therefore, the following function is continuous at
This means that is a removable discontinuity. Note that a removable discontinuity is typically a hole in the graph.

Non-Removable Discontinuity

When a function cannot be redefined so that the point of discontinuity becomes a valid input, it is called a non-removable discontinuity. Consider, for example, Since is not in the domain of the function, there is a point of discontinuity at

Rational function with non-removable discontinuity
This is a non-removable discontinuity because there is no way to redefine the function so that it becomes continuous at
Example

Modeling Annual Income Per Capita

LaShay lives in Des Moines, where the total annual amount of personal income in millions dollars, and the population in millions, are modeled by the following functions.
In these functions, represents the number of years that have passed since
a Write a function modeling the annual income per capita in Des Moines.
b Identify the points of discontinuity of , if any.

Hint
a Divide the total amount of personal income by the population. Recall how to divide rational expressions.
b For which values of is the function undefined?

Solution
a Take a look at what the given equations represent.
Total Annual Income Population
In order to write a function for the annual income per person the total annual income must be divided by the population This will give the annual income of a person.
SubstituteExpressions

Substitute expressions

DivFracSL

The binomials in the denominator can be multiplied.
MultPar

Multiply parentheses

AddTerms

Add terms

b Consider the function written in Part A.
Recall that division by zero is not defined. Therefore, the rational function is undefined when either of the parentheses equals zero.
is undefined
This means that and are excluded values.
If a real number is not in the domain of a function, then the function has a point of discontinuity at Since the domain is all real numbers except and there are two points of discontinuity of the function.
Example

Finding Point of Discontinuity

On the way to the pencil factory, LaShay noticed that road maintenance work is being carried out to fill a hole on the road.

A map that shows a lake, a forrest, some buildings, and a road with a hole in it
The equation below models the road to the factory.
a Find the coordinate of the point where the road maintenance is being completed.
b The municipality of Des Moines plans to build a new road, which is modeled by the following function.
Graph the function.

Answer
a
b Graph:

Hint
a Start by factoring the numerator and the denominator of the function. Find the point of discontinuity of the graph.
b Check if the numerator and the denominator have a common factor.

Solution
a The terms in the numerator of the function have a common factor of Also, the denominator can be factored by grouping.
FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

Notice that there is no value that can make equal zero. However, the other factor in the denominator makes the function undefined when Therefore, this value is excluded from the domain.
Recall that the point of discontinuity occurs at when is not in the domain of a function. Consequently, there is a point of discontinuity at
In the context of the problem, this is the coordinate of the point where the road is being repaired.

Notice that it is also a removable discontinuity because is the common factor in the numerator and denominator.

b Start by factoring the numerator and denominator to see if the function can be simplified.
▼
Factor
WriteDiff

Write as a difference

FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

The function is undefined when This means that the domain of the function is all real numbers except After simplification, the function will be a linear function except at
Note that Therefore, the graph of the given function is the graph of the function with a hole at
The graph of N(x)

This graph also has a removable discontinuity at because is the common factor in the numerator and denominator of the rational function.

Discussion

Asymptotes of Rational Functions

An asymptote of a graph is an imaginary line that the graph gets close to as goes to plus or minus infinity or a particular number. For example, the graph of the rational function has two asymptotes — the axis and the axis.

Graph of f(x)= 1/x and its asymptotes

Analyzing the diagram, the following can be observed.

  • As approaches infinity and as approaches negative infinity, the value of the function approaches Therefore, or the axis, is a horizontal asymptote of the graph of
  • As approaches the value of the function approaches either positive or negative infinity. Therefore, or the axis, is a vertical asymptote of the graph of
In the coordinate plane below, the asymptotes for three different graphs are shown.
The graphs with asymptotes of three different functions
The applet demonstrates that asymptotes can be not only vertical and horizontal, but also oblique.
 Consider a rational function where is a polynomial with a leading coefficient of and a degree of and where is a polynomial with a leading coefficient of and a degree of
To determine the asymptotes of the function, the denominator and the degrees of the polynomials in the numerator and denominator must be analyzed.

Identifying Vertical Asymptotes

To identify the vertical asymptotes, the function should be in its simplest form. That is, if and have common factors, the function should be simplified first. The vertical asymptotes will occur at the zeros of the denominator. Check out the following examples.

Identifying vertical asymptotes

Identifying Horizontal Asymptotes

The degrees of polynomials in the numerator and denominator of a rational function will determine if the graph has a horizontal asymptote.

Horizontal Asymptote If there is no horizontal asymptote.
If the horizontal asymptote is
If the horizontal asymptote is
The graph of a rational function may have one or more vertical asymptotes and at most one horizontal asymptote.
Functions y=x/(x^2-9), y=3x^2/(x^2-x), and y=-x^4/(5x^2-5) are graphed with the vertical and horizontal asymptotes
Notice that the graph of a rational function will never cross a vertical asymptote, whereas the graph may or may not cross a horizontal asymptote.
Example

Identifying Asymptotes

LaShay takes a travel-sickness pill about minutes before the school bus leaves for the field trip. The amount in milligrams of the medication in her bloodstream is modeled by the following rational function.
Here, represents the time in hours after one pill is taken. Identify the asymptotes of the function.

Answer

Vertical Asymptotes: and
Horizontal Asymptote:

Hint

Factor the numerator. What does the Factor Theorem state?

Solution
To identify the asymptotes, the numerator and denominator of the function need to be factored. Notice that the terms in the numerator have a common factor of
FactorOut

Factor out

To factor the polynomial in the denominator, use the Factor Theorem.

Factor Theorem

The expression is a factor of a polynomial if and only if the value is a zero of the related polynomial function.

Now, the aim is to find all real zeros of the polynomial. Before doing so, identify the degree of the polynomial.
It is a polynomial with a degree of Subsequently, by the Fundamental Theorem of Algebra, it is known know that has exactly three roots. Additionally, by the Rational Root Theorem, integer roots must be factors of the constant term. Since the constant term of is the possible integer roots are and
Three roots for were found, so there is no need for further investigation. In conclusion, and are all the real roots of Next, apply the Factor Theorem to write its factored form.
Now that the factored form of the denominator has been found, the given function can be simplified.
Substitute

CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

Since the numerator and the denominator share the factor there is a hole at Also, the denominator becomes zero when and These values make the function undefined. Therefore, the function has two vertical asymptotes.
To find the horizontal asymptote, compare the degrees of the polynomials in the numerator and denominator.
Asymptote
none
Look at the degrees of the numerator and denominator of the function.
The degree of the denominator is greater than the degree of the numerator, Therefore, the horizontal asymptote is
Discussion

Oblique Asymptote

An asymptote can be neither vertical nor horizontal. It can be a slanted line. In such cases, it is called an oblique asymptote or slant asymptote.

Graph of f(x)=(x^2+1)/x with a vertical and oblique asymptotes
An oblique asymptote of a rational function occurs when the degree of the numerator is more than that of the denominator. Therefore, a function with an oblique asymptote can never have a horizontal asymptote. To find the equation of the oblique asymptote of a rational function is divided by using long division to get a quotient with a remainder
The equation of the oblique asymptote is of the form where is a non-zero real number.

Example

To illustrate how the technique is used, the asymptotes of the following function will be found.
The difference between the degrees of the numerator and denominator is This means that its graph does not have a horizontal asymptote but an oblique asymptote. To find the oblique asymptote, polynomial long division will be used.
▼
Divide

Multiply by

Subtract down

The quotient is with a remainder of
Therefore, is the oblique asymptote of the rational function.
Graph of f(x) = (x^3+2x)/(4x^2-1)
Example

Modeling Interest in The Trip

The students spent about hours at the factory. At the end of this period, LaShay's teacher made a survey about the trip and drew a graph relating the students' interest in the trip and the time elapsed at the factory. The greater the value, the greater the interest of the students.

The graph of the function S(t) in the first quadrant
The graph above is the part of the graph of the following function in the first quadrant.
Identify the asymptotes of the function.

Answer

Vertical Asymptotes:
Oblique Asymptote:

Hint

Start by subtracting the rational expressions on the right-hand side of the function.

Solution
First, the rational expressions on the right-hand side will be subtracted. The least common denominator of the expressions is Therefore, the first expression needs to be multiplied by so that both have the same denominator.
▼
Simplify right-hand side
ExpandFrac

Multiply

Multiply

SubFrac

Subtract fractions

Distr

Distribute

CommutativePropAdd

Commutative Property of Addition

▼
Rewrite the denominator
The numerator and the denominator do not have any common factors. Therefore, the function does not have a point of discontinuity, or hole, in its graph. However, it has a vertical asymptote at because the function is undefined at that value.
Now, to determine if the given rational function has a horizontal or oblique asymptote, the degrees of the polynomials in the numerator and the denominator will be compared.
The degree of the numerator is greater than the degree of the denominator, so the graph has no horizontal asymptotes. However, since the degree of the numerator is more than that of the denominator, the graph has an oblique asymptote. Its equation is found by dividing the numerator by the denominator. Use long division to do this.
▼
Divide

Multiply by

Subtract down

▼
Divide

Multiply by

Subtract down

The given function can be rewritten as follows.
Therefore, the equation of the oblique asymptote is
Discussion

Graphing Rational Functions

Rational functions may have more than one vertical asymptote, which can make graphing them a bit more difficult than graphing reciprocal functions. However, the asymptotes are useful when graphing a rational function because they can describe the end behavior. Consider graphing the following function.
To graph the rational function, these four steps can be followed.
1
Draw the Asymptotes
expand_more
First, the vertical asymptote(s) of the function will be found. If the numerator and the denominator of a rational function have no common factors, then the vertical asymptote is the value that makes the denominator zero. Start by factoring the denominator to see candidates for asymptotes.
▼
Factor the denominator
WriteDiff

Write as a difference

FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

The rational function is undefined where and This means that and are not included in the domain. Notice that is a common factor between the numerator and the denominator. Next, check if there are any other common factors and cancel them.
▼
Factor the numerator
WriteDiff

Write as a difference

FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

▼
Simplify
CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

Since the factor was canceled, the graph has a hole at Also, the graph has a vertical asymptote at because is not included in the domain.
The degrees of the polynomials in the numerator and denominator determine whether the function has a horizontal or oblique asymptote. In the following table, and are the degrees of the polynomials in the numerator and denominator, and and are their respective leading coefficients.
Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique
Now consider the degrees of the numerator and denominator for the given function. The simplified form of the function can also be considered at this point because both have the same graph.
The degree of the numerator minus the degree of the denominator equals Therefore, the function has an oblique asymptote. The equation of the oblique asymptote is found by dividing the numerator by the denominator without considering the remainder. To divide the polynomials, long division will be used.
▼
Divide

Multiply by

Subtract down

▼
Divide

Multiply by

Subtract down

The equation of the oblique asymptote is Finally, draw the asymptotes on a coordinate plane.
Vertical asymptote x=4 and oblique asymptote y=x+9

The given function has one vertical asymptote. Therefore, its graph will consist of two parts, one to the left and the other to the right of the vertical asymptote. These parts may lie above or below the oblique asymptote.

2
Find the Intercepts
expand_more
The intercepts are the points where the graph intersects the axis. The value of the coordinate is zero at these points. Substitute for in the given function and solve for The simplified function can be used to find the zeros.
Substitute

▼
Solve for
MultEqn

ZeroProdProp

Use the Zero Product Property

AddEqn

SubEqn

The graph has two intercepts. The intercept is the point where the graph intersects the axis. At this point, the value of the coordinate is zero.
Substitute

▼
Evaluate right-hand side
IdPropAdd

Identity Property of Addition

Multiply

Multiply

CalcQuot

Calculate quotient

There is a intercept at Show the intercepts on the same coordinate plane.
The asymptotes and intercepts on the same coordinate plane
3
Make a Table of Values
expand_more

Some additional points are needed to get a good sense of the shape of the graph. Make sure to only use values included in the domain of the function.

It is almost done. Plot the points and imagine how the shape of the graph should look!

Asymptotes, intercepts and points from table of values on a coordinate plane

As can be seen, one part of the graph will lie to the left of the vertical asymptote and below the oblique asymptote. The other part of the graph will be to the right of the vertical asymptote and above the oblique asymptote.

4
Draw the Graph
expand_more

The graph can now be drawn by connecting the points with a smooth curve. It must approach the asymptotes. Do not forget to plot the hole at

The graph of the function
Example

Graphing the Function for the Surface Area to Volume Ratio of a Box

The factory sells pencils in a rectangular box. The dimensions of the box are shown in the diagram.

6 pencils in a rectangular box height 2x+6, width x-2, depth 2
a Write a function of the form where and are the surface area and volume of the box, respectively.
b Graph the function

Answer
a
b Graph:
The graph of the function E(x)

Hint
a The surface area of a prism is the sum of the areas of its surfaces. The volume of a prism is the product of its dimensions.
b Start by determining the vertical and horizontal asymptotes. Then, make a table of values.

Solution
a First, the surface area and volume of the given box will be found.
Rectangular box with dimensions, (2x+6), x, and (x-2)
The surface area of a rectangular box is the total area of all the surfaces of the shape. Since the dimensions are given in terms of the surface area is a function of
▼
Simplify right-hand side
Multiply

Multiply

Distr

Distribute

Distr

Distribute

MultPar

Multiply parentheses

AddSubTerms

Add and subtract terms

Next, the volume of the pencil box will be calculated. Recall that the volume of a rectangular prism is the product of its dimensions.
▼
Simplify right-hand side
Distr

Distribute

MultPar

Multiply parentheses

AddTerms

Add terms

Finally, the function the ratio of to can be written.
SubstituteExpressions

Substitute expressions

▼
Simplify right-hand side
b To graph the given rational function, its asymptotes and intercepts will be found. A table of values will then be used to draw the graph.

Asymptotes

Consider the function written in the previous part.
First, factor the denominator of this function to identify its domain.
WriteDiff

Write as a difference

▼
Factor out
FactorOut

Factor out

FactorOut

Factor out

FactorOut

Factor out

The rational function is undefined where and because division by zero is not defined. This means that and are not included in the domain.
Asymptotes can be vertical, horizontal, or slanted lines. Once again, consider the function.
It seems that the denominator and the numerator do not have a common factor. To find the factors of the numerator, its zeros can be found using the Quadratic Formula.
As shown, they do not have common factors. Therefore, there are no holes in the graph. Recall that if the real number is not included in the domain, there is a vertical asymptote at In this case, there are two vertical asymptotes, one at and the other at
To find whether the graph has a horizontal or oblique asymptote, review how its type is determined using the degrees of the polynomials in the numerator and the denominator.
Asymptote
none
the quotient of the polynomials with no remainder
Look at the degrees of the numerator and denominator of the function.
The degree of the denominator is equal to the degree of the numerator. Therefore, the line is the horizontal asymptote of the function.
Vertical asymptotes x=-3 and x=2, a horizontal asymptote y=1

Intercepts

To find the intercepts, is substituted for into the function. It will be better to use the form in which the numerator and the denominator are factored.
Substitute

▼
Solve for
MultEqn

ZeroProdProp

Use the Zero Product Property

SubEqn

SubEqn

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

To find the intercept, substitute for in the function and solve for This time the form of the function found in Part A can be used.
Substitute

▼
Solve for
CalcPow

Calculate power

ZeroPropMult

Zero Property of Multiplication

IdPropAdd

Identity Property of Addition

ReduceFrac

There is a intercept at Finally, all the intercepts can be plotted on the coordinate plane.
Asymptotes and intercepts (-2-\sqrt{6},0), (-2+\sqrt{6},0), and (0,1/3) are graphed on the coordinate plane

Table of Values

Now, make a table of values to graph the given function.

Finally, the graph of the function can be drawn by plotting the found points and connecting them with a smooth curve. Recall that a rational function can cross the horizontal asymptote but cannot cross the vertical asymptotes. In this case, the horizontal asymptote is crossed.

The graph of the function E(x)

Extra

 Factoring The Numerator of

Recall the Factor Theorem.

has a factor if and only if

In other words, the polynomial function has zeros at the values. Since the numerator of is a quadratic expression its zeros can be found by using the Quadratic Formula.
The values of and of the numerator are and respectively. Substitute these values into the formula.
SubstituteValues

Substitute values

▼
Evaluate right-hand side
IdPropMult

Identity Property of Multiplication

CalcPow

Calculate power

MultNegNegOnePar

AddTerms

Add terms

SplitIntoFactors

Split into factors

SqrtProd

CalcRoot

Calculate root

FactorOut

Factor out

SimpQuot

Simplify quotient

The zeros of the numerator are and By the Factor Theorem, the following is the factored form of the numerator.
Example

Modeling the Amount of Water Pumped Into the Factory

Factory managers plan to pump water into the factory from a pond near the factory. The managers modeled the amount of water pumped in thousands of barrels per year years after pumping starts to be as follows.
a Use a graphing calculator to find the intercepts rounded to the nearest integer. Interpret the intercepts.
b Determine if the graph has horizontal or oblique asymptotes. If the graph has an oblique asymptote, find its equation.

Answer
a $\bm{x}\textbf{-intercept:}$

Interpretation: In about years, there will be no water left in the pond.

b The graph has an oblique asymptote because the difference between the degrees of the numerator and the denominator is

Equation:

Hint
a Use the zero option in the calculator. It may be necessary to re-size the viewing window in order to see the entire graph.
b Compare the degrees of the numerator and the denominator.

Solution
a In order to find any intercepts, or zeros, of the function, look at the graph of the function using a graphing calculator. To draw the function on the calculator, push the button and type the equation in the first row. 
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Now, push to draw the function. 

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It appears that only one part of the graph is shown, but it is not possible to know that until the size of the viewing window is changed. 

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The graph does not seem to have any zeros. However, note that as increases in the first quadrant, the graph gets closer to the axis and could cross it. To check this, zoom in on this part by changing the window settings. Push and change the settings as shown below. Then push once more to draw the equation with these new settings. 

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Now it can be seen that the graph intersects the axis and there is a zero. To find it, use the zero option in the calculator. It can be found by pressing and then 

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After selecting the zero option, choose left and right boundaries for the zero. Finally, the calculator asks for a guess where the zero might be. After that, it will calculate the exact point. 

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The function intersects the axis at about Since represents the numbers of years that have passed since pumping starts, the intercept means that in about years, there will be no water left in the pond.

b Recall the table used to determine what type of asymptote a rational function can have.
Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique
Now, compare the degrees of the numerator and denominator of the given function.
The degree of the numerator minus the degree of the denominator equals Therefore, the function has an oblique asymptote. The equation of this asymptote is found by dividing the numerator by the denominator and disregarding any remainder. To divide the polynomials, use long division.
▼
Divide

Multiply by

Subtract down

▼
Divide

Multiply by

Subtract down

The given function can be rewritten as follows.
Therefore, the equation of the oblique asymptote is
Closure

Graphing the Average Cost Function

Rational functions can have more than one vertical asymptote but at most one horizontal or oblique asymptote. Also, if the numerator and denominator have a common factor then there is a hole at With these tips in mind, the graph of the function that LaShay wrote at the beginning of the lesson can now be graphed.
The function gives the average cost in thousands dollars for producing thousand pencils.
a Graph the average cost function.
b Identify the domain and range of the function.

Answer
a Graph:
The graph of the function A(x)
b Domain:
Range:

Hint
a Start by identifying asymptotes. Then, make a table of values using values that are around the vertical asymptote.
b Examine the graph and see if the graph is symmetric about the point of intersection of the asymptotes. Determine all the values for which the function is graphed.

Solution
a The steps to draw a rational function can be listed as follows.
  1. Draw the asymptotes.
  2. Find any intercepts.
  3. Make a table of values.
  4. Draw the graph.

Drawing the Asymptotes

The vertical asymptotes of a rational function are the values where the denominator of the function is zero.
The denominator is when This means that this value is not included in the domain.
Since the numerator and denominator do not share a common factor, the graph does not contain any holes. Additionally, the graph has a vertical asymptote at the value excluded from the domain.
Next, recall the table for determining the other asymptotes.
Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique
Examine the degrees of the numerator and denominator.
The degree of the numerator minus the degree of the denominator equals Therefore, the function has an oblique asymptote. To find its equation, divide the numerator by the denominator using long division.
▼
Divide

Multiply by

Subtract down

▼
Divide

Multiply by

Subtract down

The equation of the oblique asymptote is
The asymptotes can be drawn on a coordinate plane.
The asymptotes x=0 and y=0.2x+5 are graphed on a coordinate plane

Finding the Intercepts

The intercept and intercept are the points where a graph crosses the and axes, respectively. Subsequently, the intercepts occurs when
Substitute

MultEqn

▼
Solve using the quadratic formula
UseQuadForm

Use the Quadratic Formula:

CalcPowProd

Calculate power and product

SubTerm

Subtract term

Notice that there is a negative number under the radical symbol. Therefore, the function has no real solutions. In other words, its graph does not cross the axis. Furthermore, the graph has no intercept as the domain does not contain

Making a Table of Values

A table of values will be used to get a rough idea of the shape of the graph.

Plot the points and imagine how the shape of the graph should look!

The points found in the table are plotted

Finally, draw the graph by connecting the points. It must approach, but not cross, the asymptotes.

The points are connected with a smooth curve
b The domain of the function was identified when graphing the function.
To identify the range of the function, take a look at its graph.
The graph of the function A(x) with a highlighted region between the lower and upper parts of the graphs

As shown, the range does not contain all real numbers. It appears that the minimum point of the part of the graph in the first quadrant is Notice also that the graph is symmetric about the point of intersection of the asymptotes, Using this fact, the maximum point of the other part can be identified.

The graph of the function A(x) with points (50,25) and (-50,-15)
It is the point Therefore, the range is all the values less than or equal to and greater than or equal to

Checking Our Answer

 Use a Graphing Calculator

A graphing calculator can be used to check the answers. The graph of the function will be drawn first. Press the button and type the equation in the first row. 

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By pushing the calculator will draw the equation. For this function to be visible on the screen, re-size the standard window by pushing the button. Change the settings to a more appropriate size and then push 

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Next, the local maximum and local minimum will be found. To do so, push then and choose the maximum option. 

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When using the maximum feature, choose the left and right bounds. The calculator will then provide a best guess as to where the maximum might be. 

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The maximum point of the graph in the third quadrant is To find the minimum, repeat the same process but choose the minimum option. 

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The minimum point of the part in the first quadrant is As a result, the range is and the domain is all real numbers except


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