Chapter 1.7, Problem 31E

(a)

To determine

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

Explanation of Solution

Given information:

Consider $g(x)=2[[x]]-1$

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

We have $g(x)=2[[x]]-1$

The function $f(x)=[[x]]$ is the simplest function and have same shape as that of $g(x).$

Hence, the parent function is $f(x)=[[x]]$ .

(b)

To determine

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

Explanation of Solution

Given information:

Consider $g(x)=2[[x]]-1$

The parent function is $f(x)=[[x]]$ .

If the function $f(x)=[[x]]$ is vertically stretched by the factor $2$ and right shift one unit ,

then $g(x)=2[[x]]-1$ is obtained.

Hence, the transformation of $f\text{ to }g$ is vertical stretch and then right shift one unit.

(c)

To determine

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

Explanation of Solution

Given information:

Consider $g(x)=2[[x]]-1$

Graph:

The graph of $g(x)=2[[x]]-1$

(d)

To determine

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

Explanation of Solution

Given information:

Consider $g(x)=2[[x]]-1$

Parent function is $f(x)=[[x]]$

We have

  $\begin{array}{l}g(x)=2[[x]]-1\\ =2f(x)-1\end{array}$

Hence, the parent function is $g(x)=2f(x)-1$ .