Chapter 1.7, Problem 31E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 31E

For the function $g(x)={(x-1)}^3+2$ , the parent function is $f(x)={(x-1)}^3$ .

Explanation of Solution

Given information: $g(x)={(x-1)}^3+2$

Concept used:

To identify the parent function, strip all the arithmetic operations on the function to leave behind one higher order operation in just x.

Conclusion:

So, remove the arithmetic operation of addition by 2 from the given funcion to get the parent function.

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 31E

The shape of 1y13y $f(x)={(x-1)}^3$ is drawn reflected in the x-axis and then shifted by 2 units.

Explanation of Solution

Given information: $g(x)={(x-1)}^3+2$ and

  $f(x)={(x-1)}^3$

Concept Used:

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the Parent function f to g .

Conclusion:

The shape of $f(x)={(x-1)}^3$ is drawn reflected in the x-axis and then shifted by 2 units is the required sequence of transformations from $f$ to $g$ .

(c)

To determine

To sketch the graph of $g$ .

Answer to Problem 31E

  

Explanation of Solution

Given information : $g(x)={(x-1)}^3+2$

Concept Used :

Use the sequence of transformation to plot the graph of the function

Conclusion:

Obtain the graph of $g(x)={(x-1)}^3+2$

(d)

To determine

To write $g$ in terms of $f$

Answer to Problem 31E

  $g(x)=f(x)+2$

Explanation of Solution

Given information:

  $g(x)={(x-1)}^3+2$

  $f(x)={(x-1)}^3$

Conclusion:

Add 2 to $f(x)$ to get $g(x)$ in terms of $f(x)$

  $g(x)=f(x)+2$