Chapter 1.4, Problem 56E

a.

To determine

Identify the parent function f for the given function.

Answer to Problem 56E

Parent function is $f(x)=x^3$

Explanation of Solution

Given:

  $g(x)=\frac{1}{2}{(x+1)}^3$

Calculation:

To identify the parent function of given function, first $g(x)$ has to be plotted.

Calculation for graph:

Consider, $g(x)=\frac{1}{2}{(x+1)}^3$

Values of x	Values of g (x )
-2	-0.5
-1	0
0	0.5
1	4
2	13.5

By taking different values of x , the graph can be plotted.

Graph:

  

Interpretation:

From the above graph, it is clear that, the parent function is: $f(x)=x^3$ .

Conclusion:

Therefore, the parent function is: $f(x)=x^3$

b.

To determine

To determine the transformation from f to g.

Explanation of Solution

Given:

  $g(x)=\frac{1}{2}{(x+1)}^3$

Calculation for graph:

Consider, $g(x)=\frac{1}{2}{(x+1)}^3$

Values of x	Values of g (x )
-2	-0.5
-1	0
0	0.5
1	4
2	13.5

By taking different values of x , the graph can be plotted.

Graph:

  

Calculation for graph:

Consider $f(x)=x^3$

Values of x	Values of f (x )
0	0
1	1
-1	-1
2	8
-2	-8

By taking different values of x , the graph can be plotted.

Graph:

  

Now graph the functions $f(x)$ and $g(x)$ together.

  

Interpretation:

Below are the steps to transform $f(x)$ to $g(x)$ .

Step 1: Shift the base graph by 1 units to the left.

  

Step 2: Then vertically shrink the obtained graph by (1/2) units.

  

Conclusion:

So, by following the above steps, the graph of $f(x)$ can be transformed to $g(x)$ .

c.

To determine

To sketch the graph of the function $g(x)=\frac{1}{2}{(x+1)}^3.$

Explanation of Solution

Given:

  $g(x)=\frac{1}{2}{(x+1)}^3$

Calculation for graph:

Consider, $g(x)=\frac{1}{2}{(x+1)}^3$

Values of x	Values of g (x )
-2	-0.5
-1	0
0	0.5
1	4
2	13.5

By taking different values of x , the graph can be plotted.

Graph:

  

Interpretation:

The above graph represents the given function $g(x)$ .

d.

To determine

To write $g(x)$ as a function of parent function $f(x)$

Answer to Problem 56E

  $g(x)=\frac{1}{2}f(x+1)$

Explanation of Solution

The parent function is: $f(x)=x^3$

Hence, $f(x+1)={(x+1)}^3$ -----------------------------(i)

Given,

  $g(x)=\frac{1}{2}{(x+1)}^3$

Putting the value of (i) in $g(x)$

  $g(x)=\frac{1}{2}f(x+1)$

Conclusion:

Hence, the value of $g(x)$ in terms of $f(x)$ is $g(x)=\frac{1}{2}f(x+1)$ .