Chapter 1.7, Problem 78E

(a)

To determine

To find: The intervals on which the function $y=f\left(-x\right)$ is increasing or decreasing.

Answer to Problem 78E

The intervals are given below.

Explanation of Solution

Given:

The given graph is as:

  

Calculation:

The graph is reflected over $y$ axis. Reflect the specified intervals over the $y$ axis. Increasing intervals will now become decreasing intervals, and decreasing intervals will now become increasing intervals.

So, the interval for increasing graph is $\left(-2,1\right)$ & interval for decreasing graph is $\left(1,\infty \right),\left(-\infty ,-2\right)$ .

(b)

To determine

To find: The intervals on which the function $y=-f\left(x\right)$ is increasing or decreasing

Answer to Problem 78E

The intervals are given below.

Explanation of Solution

The graph is reflected over the $x$ axis. Increasing intervals now become decreasing and decreasing intervals will now become increasing intervals.

The interval for increasing graph is $\left(-1,2\right)$

The interval for decreasing graph is $\left(-\infty ,1\right),\left(2,\infty \right)$ .

(c)

To determine

To find: The intervals on which the function $y=\frac{1}{2}f\left(x\right)$ is increasing or decreasing.

Answer to Problem 78E

The intervals are given below.

Explanation of Solution

Calculation:

There is a vertical shrink by $\frac{1}{2}$ . All intervals will remain the same.

So the interval for increasing graph is $\left(1,\infty \right),\left(-\infty ,-2\right)$ .

The interval for decreasing graph is $\left(-1,2\right)$ .

(d)

To determine

To find: The intervals on which the function $y=-f\left(x-1\right)$ is increasing or decreasing.

Answer to Problem 78E

The intervals are given below.

Explanation of Solution

The graph is reflected over the $x$ axis. Shift the intervals right by $1$ . Increasing intervals will now become decreasing intervals, and decreasing intervals will now become increasing intervals.

So the interval for increasing graph is $\left(0,3\right)$ .

The interval for decreasing graph is $\left(-\infty ,2\right),\left(3,\infty \right)$ .

(e)

To determine

To find: The intervals on which the function $y=f\left(x-2\right)+1$ is increasing or decreasing.

Answer to Problem 78E

The intervals are given below.

Explanation of Solution

In the given graph shift the intervals by $2$ . Everything else remains the same.

So the interval for increasing graph is $\left(-\infty ,1\right),\left(4,\infty \right)$ .

The interval for decreasing graph is $\left(1,4\right)$ .