Chapter 1.7, Problem 70E

To determine

To Check: whether the obtained graph is same after reversing the order of transformation in 2(a) and 2(b)

Answer to Problem 70E

Graph for $-x^4-2$ will not be same and graph for $-{\left(x-3\right)}^4$ will be same.

Explanation of Solution

Given information:

From 2(a) $g\left(x\right)=-x^4+2$

From 2(b) $g\left(x\right)=-{\left(x-3\right)}^4$

Concept used:

The parent function is $f\left(x\right)=x^4$

Calculation:

For $g\left(x\right)=-x^4+2$

It is obtained by reflecting the parent function about x- axis and shifting upward by two units.

The transformed function is $y=-x^4-2$

Now on reversing the steps a new transformed function $g\left(x\right)$ is obtained

Vertical shift of the parent function by 2 units upward gives $g\left(x\right)=x^4+2$

Reflecting about x- axis gives $g\left(x\right)=-x^4-2$

But $-x^4-2\ne -x^4+2$

This implies that the graphs are not same on reversing the order of transformation.

For $g\left(x\right)=-{\left(x-3\right)}^4$

It is obtained by Horizontal shift by 3 units right of the parent function and then reflecting about x- axis.

The transformed function is $y=-{\left(x-3\right)}^4$

Now on reversing the steps a new transformed function $g\left(x\right)$ is obtained

Reflecting the parent function about x- axis gives $g\left(x\right)=-x^4$

Horizontal shift of $g\left(x\right)=-x^4$ by 3 units right gives $g\left(x\right)=-{\left(x-3\right)}^4$

Here $-{\left(x-3\right)}^4=-{\left(x-3\right)}^4$

This implies that the graphs are same on reversing the order of transformation.