Chapter 2.2, Problem 105E

a.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=f\left(x\right)+2$

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)=x^4+2$

Let us put $-x$ in the function $g\left(x\right)$ .

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

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

Compairing the graphs of functions $g\left(x\right)$ and $f\left(x\right)$ , the graph of the function $f\left(x\right)$ is vertical shift by $2$ units upward.

Hence, the graph of the function $g\left(x\right)$ is even and vertical shift by $2$ units upward than $f\left(x\right)$ .

b.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=f\left(x+2\right)$

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)={(x+2)}^4$

Let us put $-x$ in the function $g\left(x\right)$ .

  $g\left(-x\right)={(-x+2)}^4$

So, the function $g\left(x\right)$ is neither even nor odd.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

Compairing the graphs of functions $g\left(x\right)$ and $f\left(x\right)$ , the graph of the function $f\left(x\right)$ is vertical shift by $2$ units to the left.

Hence, the graph of the function $g\left(x\right)$ is even and vertical shift by $2$ units to the left than $f\left(x\right)$ .

c.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g(x)=f(-x)$

Putting the function $f\left(x\right)$ in the equation

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

Let us put $-x$ in the function $g\left(x\right)$ .

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

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

The graph of the function $g\left(x\right)$ is in reflection of $y-axis$ than graph of $f\left(x\right)$ .

Hence, the graph of the function $g\left(x\right)$ is even and is in reflection of $y-axis$ than than $f\left(x\right)$ .

d.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=-f\left(x\right)$

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)=-x^4$

Let us put $-x$ in the function $g\left(x\right)$ .

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

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

The graph of the function $g\left(x\right)$ is in reflection of $x-axis$ than graph of $f\left(x\right)$ .

Hence, the graph of the function $g\left(x\right)$ is even and is in reflection of $x-axis$ than than $f\left(x\right)$ .

e.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=f\left(\frac{1}{2}x\right)$

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)={\left(\frac{1}{2}x\right)}^4$

Let us put $-x$ in the function $g\left(x\right)$ .

  $\begin{array}{l}g\left(-x\right)={\left(-\frac{1}{2}x\right)}^4\\ g\left(-x\right)={\left(\frac{1}{2}x\right)}^4\\ g\left(-x\right)=g\left(x\right)\end{array}$

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

Compairing the graphs of functions $g\left(x\right)$ and $f\left(x\right)$ , the graph of the function $g\left(x\right)$ is horizontal stretch.

Hence, the graph of the function $g\left(x\right)$ is even and horizontally stretched.

f.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g(x)=\frac{1}{2}f\left(x\right)$

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)=x^4+2$

Let us put $-x$ in the function $g\left(x\right)$ .

  $\begin{array}{l}g\left(-x\right)=\frac{1}{2}(-x^4)\\ g\left(-x\right)=\frac{1}{2}{x}^4\\ g\left(-x\right)=g\left(x\right)\end{array}$

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

Compairing the graphs of functions $g\left(x\right)$ and $f\left(x\right)$ , the graph $g\left(x\right)$ is vertical shrink.

Hence, the graph of the function $g\left(x\right)$ is even and vertically shrink.

g.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=f(x^{\frac{3}{4}})$ for $x\ge 0$ .

Putting the function $f\left(x\right)$ in the equation

  $g\left(x\right)=x^3$

Let us put $-x$ in the function $g\left(x\right)$ .

  $\begin{array}{l}g\left(-x\right)=(-x^3)\\ g\left(-x\right)=-x^3\\ g\left(-x\right)=-g\left(x\right)\end{array}$

So, the function $g\left(x\right)$ is neither odd nor even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

  

Hence, the graph of the function $g\left(x\right)$ is neither even nor odd.

h.

To determine

Sketch the graph of the function.

Answer to Problem 105E

  

Explanation of Solution

Calculation:

Let us draw a graph of $f\left(x\right)=x^4$

  

The given function is,

  $g\left(x\right)=fof\left(x\right)$

Putting the function $f\left(x\right)$ in the equation

  $\begin{array}{l}g\left(x\right)={(}^{x^4}\\ g\left(x\right)=x^{16}\end{array}$

Let us put $-x$ in the function $g\left(x\right)$ .

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

So, the function $g\left(x\right)$ is even.

Let us draw a graph of $g\left(x\right)$ and $f\left(x\right)$ function .

Hence, the graph of the function $g\left(x\right)$ is even.