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Chapter 4: Q. 36 (page 245)

Discuss each rational function following Steps 1鈥�7 given on page 228.

F(x)=x3x2-4

Short Answer

Expert verified

The domain is x|x鈮�-2,2and graph of the function is

Step by step solution

01

Step 1. Given Information 

The given function isF(x)=x3x2-4

02

Step 2. Explanation 

Factor the numerator and denominator.

F(x)=x3(x+2)(x-2)

The domain isx|x鈮�-2,2

Since 0 is in the domain, find the y-intercept.

F(0)=03(0+2)(0-2)=0-4=0

Thus, the y-intercept is 0

The real zero of the numerator satisfies the equationx3=0

Hence, the only x-intercept is 0.

03

Step 3. Explanation  

Determine the behavior of the graph near the x-intercept by substituting 0 in the denominator.

F(x)=x3(0+2)(0-2)=x32(-2)鈮�-14x3

Plot the point (0,0)and indicate a line with slope -14on the graph.

The real zeroes of the denominator are the real solutions of the equation (x+2)(x-2)=0

The two lines x=-2,x=2are the two vertical asymptotes of the graph.

The degree of numerator is greater than the degree of denominator. So, the function is improper. Use the long division to find the horizontal asymptote.

Thus, F(x)=x+4x(x+2)(x-2)

As the quotient is x. So, the line y=xis an oblique asymptote of the graph.

04

Step 4. Calculation  

Substitute F(x)=xto determine whether the graph intersects the asymptote y=x

x3x2-4=xx3=x3-4x0=-4x0=x

Thus, the point of intersection is (0,0)

Construct the table and determine the values in the four intervals.

05

Step 5. Graph  

Plot the points and lines on the graph.

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Most popular questions from this chapter

In Problems 63鈥�72, find the real solutions of each equation.

3x3-x2-15x+5=0

In Problems 57鈥� 62, find the real zeros of f. If necessary, round to two decimal places.

f(x)=x3+3.2x2-16.83x-5.31

In Problems 21鈥�32, determine the maximum number of real zeros that each polynomial function may have. Then list the potential rational zeros of each polynomial function. Do not attempt to find the zeros.

f(x)=-4x3-x2+x+2

Solve the inequality algebraically.

x4<9x2

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

F(x)=x4-16x2-2x
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