Chapter 4, Problem 4CRT

a.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x+3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $f\left(x\right)=x^3-8x$

Check whether function is even or odd.

  $\begin{array}{l}f\left(x\right)=x^3-8x\\ put,-x,in,f\left(x\right)\\ f\left(-x\right)={\left(-x\right)}^3-8\left(-x\right)\\ =-x^3+8x\\ =-\left(x^3-8x\right)\\ f\left(-x\right)=\boxed{-f\left(x\right)}\end{array}$

The function is odd. So the graph will symmetrical with respect to the origin.

The function is three degree, and leading coefficient is positive. The end behaviour of the function is,

  $y\to \infty$ as $x\to \infty$ and $y=-\infty$ as $x\to -\infty$

The graph of the function is,

  

Hence, the graph is as shown.

b.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x+3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $g\left(x\right)=-x^4+8x^2$

Check whether function is even or odd.

  $\begin{array}{l}g\left(x\right)=-x^4+8x^2\\ g\left(-x\right)=-\left(-x^4\right)+8\left(-x^2\right)\\ g\left(-x\right)=-x^4+8x^2\\ g\left(-x\right)=g\left(x\right)\end{array}$

The function is even. So the graph will symmetrical with respect to the $y-axis$ .

The function is one of the only fourdegrees, and leading coefficient is negative. The end behaviour of the function is,

  $y=-\infty$ as $x\to \infty$ and $y=-\infty$ as $x\to -\infty$

The graph of the function is,

  

Hence, the graph is as shown.

c.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x+3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $r\left(x\right)=\frac{2x+3}{x^2-9}$

The function is quotient check whether function has vertical asymptote when denominator is equal to zero.

  $\begin{array}{l}{x}^2-9=0\\ x^2=9\\ x=\pm 3\end{array}$

The graph of the function is,

  

Hence, the graph is as shown.

d.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x-3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $s\left(x\right)=\frac{2x-3}{x^2+9}$

The function is quotient check whether function has no vertical asymptote because denominator is not equal to zero. Now check for horizontal asymptote.

  $\begin{array}{l}s\left(x\right)=\frac{2x-3}{x^2+9}=\frac{x\left(2-\frac{3}{x}\right)}{x^2\left(1+\frac{9}{x^2}\right)}\\ =\frac{1\left(2-\frac{3}{x}\right)}{x\left(1+\frac{9}{x^2}\right)}\end{array}$

The factorial expression is $\frac{3}{x},\frac{1}{x},\frac{9}{x^2}$ all approaches $0$ , $x\to \pm \infty .$

  $s\left(x\right)=\frac{1\left(2-\frac{3}{x}\right)}{x\left(1+\frac{9}{x^2}\right)}\to 0.\frac{\left(2-0\right)}{\left(1+0\right)}=0$

The horizontal asymptote is the line $x=0$

The graph of the function is,

  

Hence, the graph is as shown.

e.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x-3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $h\left(x\right)=2^x-5$

The function is exponential with $5$ units shift down.

  

Hence, the graph is as shown.

f.

To determine

Match each of the following functions to the graphs.

Answer to Problem 4CRT

  

Explanation of Solution

Given information:

Without using a graphing calculator, match each of the following functions to the graphs below. Give reasons for your choices.

  $f\left(x\right)=x^3-8x$$g\left(x\right)=-x^4+8x^2$$r\left(x\right)=\frac{2x+3}{x^2-9}$

  $s\left(x\right)=\frac{2x-3}{x^2+9}$$h\left(x\right)=2^x-5$$k\left(x\right)=2^{-x}+3$

  

  

Calculation:

Consider , $k\left(x\right)=2^{-x}+3$

The function is exponential reflect it to $y-axis$ and shift it up to $3$ units.

  

Hence, the graph is as shown.