Chapter 1.7, Problem 30E

(a)

To determine

To identify: the parent function $f(x)$ of $g(x)$

Explanation of Solution

Given information:

Consider $g(x)=\frac{1}{2}\sqrt{x}$

Parent function is the simplest function of family of functions, which has the same shape as that of entire family.

We have $g(x)=\frac{1}{2}\sqrt{x}$

The function $f(x)=\sqrt{x}$ is the simplest function and have same shape as that of $g(x).$

Hence, the parent function is $f(x)=\sqrt{x}$ .

(b)

To determine

To explain: the sequence of transformations from $f(x)$ to $g(x).$

Explanation of Solution

Given information:

Consider $g(x)=\frac{1}{2}\sqrt{x}$

The parent function is $f(x)=\sqrt{x}$ .

If The function $f(x)=\sqrt{x}$ is vertically compressed by the factor $0.5$ , then $g(x)=\frac{1}{2}\sqrt{x}$ is

obtained.

Hence, the transformation of $f\text{ to }g$ is vertical compression.

(c)

To determine

To sketch: the graph of $g(x)$

Explanation of Solution

Given information:

Consider $g(x)=\frac{1}{2}\sqrt{x}$

Graph:

The graph of $g(x)=\frac{1}{2}\sqrt{x}$

(d)

To determine

To write: the function $g(x)$ in terms of $f(x)$

Explanation of Solution

Given information:

Consider $g(x)=\frac{1}{2}\sqrt{x}$

Parent function is $f(x)=\sqrt{x}$

We have

  $\begin{array}{l}g(x)=\frac{1}{2}\sqrt{x}\\ =\frac{1}{2}f(x)\end{array}$

Hence, the parent function is $g(x)=\frac{1}{2}f(x)$ .