Chapter 2.5, Problem 38AYU

a)

To determine

To find: The interval in which the graph of $y=f\left(x+2\right)$ is increasing.

Answer to Problem 38AYU

The interval in which the graph of $y=f\left(x+2\right)$ is increasingis $\left(-4,5\right)$ .

Explanation of Solution

Given:

The graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ .

Concept used:

If the graph of $y=f\left(x\right)$ is increasing over the interval $\left(a,b\right)$ thenthe interval in which the graph of $y=cf\left(x+h\right)+k$ is increasing can be found by

  $\begin{array}{l}x+h=a\\ x+h=b\end{array}$

And simplifying $x$

Calculation:

Consider the given interval and the new graph.

  $\left(-2,7\right)$ and $y=f\left(x+2\right)$

Now, let us take

  $\begin{array}{c}x+2=-2\\ x=-2-2\\ x=-4\end{array}$

Now, let us take

  $\begin{array}{c}x+2=7\\ x=7-2\\ x=5\end{array}$

Hence, the interval in which the graph of $y=f\left(x+2\right)$ is increasingis $\left(-4,5\right)$ .

b)

To determine

To find: The interval in which the graph of $y=f\left(x-5\right)$ is increasing.

Answer to Problem 38AYU

The interval in which the graph of $y=f\left(x-5\right)$ is increasingis $\left(3,12\right)$ .

Explanation of Solution

Given:

The graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ .

Concept used:

If the graph of $y=f\left(x\right)$ is increasing over the interval $\left(a,b\right)$ thenthe interval in which the graph of $y=cf\left(x+h\right)+k$ is increasing can be found by

  $\begin{array}{l}x+h=a\\ x+h=b\end{array}$

And simplifying $x$

Calculation:

Consider the given interval and the new graph.

  $\left(-2,7\right)$ and $y=f\left(x-5\right)$

Now, let us take

  $\begin{array}{c}x-5=-2\\ x=5-2\\ x=3\end{array}$

Now, let us take

  $\begin{array}{c}x-5=7\\ x=7+5\\ x=12\end{array}$

Hence, the interval in which the graph of $y=f\left(x-5\right)$ is increasingis $\left(3,12\right)$ .

c)

To determine

The interval in which the graph of $y=-f\left(x\right)$ is increasing or decreasing.

Answer to Problem 38AYU

The interval in which the graph of $y=-f\left(x\right)$ is decreasingis $\left(-2,7\right)$ .

Explanation of Solution

Given:

The graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ .

Concept used:

If the graph of $y=f\left(x\right)$ is increasing over the interval $\left(a,b\right)$ thenthe interval in which the graph of $y=cf\left(x+h\right)+k$ is increasing can be found by

  $\begin{array}{l}x+h=a\\ x+h=b\end{array}$

And simplifying $x$

Calculation:

Consider the given interval and the new graph.

  $\left(-2,7\right)$ and $y=-f\left(x\right)$

As the graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ and the graph $y=-f\left(x\right)$ can be obtained by reflecting the graph of $y=f\left(x\right)$ about x -axis.

So, the interval in which the graph of $y=f\left(x\right)$ is increasing, the graph of $y=-f\left(x\right)$ will be decreasing.

Hence, the interval in which the graph of $y=-f\left(x\right)$ is decreasingis $\left(-2,7\right)$ .

d)

To determine

The interval in which the graph of $y=f\left(-x\right)$ is increasing or decreasing.

Answer to Problem 38AYU

The interval in which the graph of $y=f\left(-x\right)$ is decreasingis $\left(-7,2\right)$ .

Explanation of Solution

Given:

The graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ .

Calculation:

Consider the given interval and the new graph.

  $\left(-2,7\right)$ and $y=f\left(-x\right)$

As the graph of $y=f\left(x\right)$ is increasing over the interval $\left(-2,7\right)$ and the graph $y=f\left(-x\right)$ can be obtained by reflecting the graph of $y=f\left(x\right)$ about y -axis.

Hence, the interval in which the graph of $y=f\left(-x\right)$ is decreasingis $\left(-7,2\right)$ .