Chapter 1.7, Problem 24E

(a)

To determine

In Exercises 21−46, g is related to one of the parent functions described in Section 1.6.

(a) Identify the parent function f

(b) Describe the sequence of transformations from f to g

(c) Sketch the graph of g

(d) Use function notation to write g in terms of f

24. g(x) = -x³-1

Identify the parent function f

Answer to Problem 24E

The Parent function is f(x)=x³

Explanation of Solution

Given:

g(x) = -x³-1

Concept Used:

A Parent function is the function in the simplest form possible for a given family of functions that tends to preserve the properties and definition of the entire family.

Conclusion:

The Parent function is f(x)=x³

(b)

To determine

Describe the sequence of transformations from f to g

Answer to Problem 24E

The shape of f(x)=x³, but reflected upon y-axis and then shifted downwards by 1 unit.

Explanation of Solution

Given:

g(x) = -x³-1

and

f(x)=x³

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)=x³, but reflected upon y-axis and then shifted downwards by 1 unit is the required sequence of transformations from f to g .

(c)

To determine

Sketch the graph of g .

Answer to Problem 24E

  

Explanation of Solution

Given:$\text{g}\left(\text{x}\right)=-{\text{x}}^3-1$

Concept Used:

We used the sequence of transformations to the plot the above graph.

Conclusion:

We thus obtain the graph for $\text{g}\left(\text{x}\right)=-{\text{x}}^3-1$

(d)

To determine

Use function notation to write $g$ in terms of $f$

Answer to Problem 24E

  $g\left(x\right)=f\left(-x\right)-1$

Explanation of Solution

Given:

  $\text{g}\left(\text{x}\right)=-{\text{x}}^3-1$

  $and$

  $f\left(x\right)=x^3$

Conclusion:

  $g\left(x\right)=f\left(-x\right)-1$