Chapter 1.4, Problem 58E

a.

To determine

Identify the parent function f for the given function.

Answer to Problem 58E

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

Explanation of Solution

Given:

  $g(x)=-{(x+3)}^3-10$

Calculation:

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

Calculation for graph:

Consider, $g(x)=-{(x+3)}^3-10$

Values of x	Values of g (x )
-2	-11
-1	-18
0	-37
1	-74
2	-135

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:

  $g(x)=-{(x+3)}^3-10$

Calculation for graph:

Consider, $g(x)=-{(x+3)}^3-10$

Values of x	Values of g (x )
-2	-11
-1	-18
0	-37
1	-74
2	-135

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 3 units to the left after taking the reflection of the base graph about x -axis.

  

Step 2: Then shift the obtained graph by 10 units vertically downwards.

  

Conclusion:

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

d.

To determine

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

Answer to Problem 58E

  $g(x)=f(x-1)+2$

Explanation of Solution

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

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

Given,

  $g(x)=-{(x+3)}^3-10$

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

  $g(x)=-f(x+3)-10$

Conclusion:

Hence, the value of $g(x)$ in terms of $f(x)$ is $g(x)=-f(x+3)-10$