Q. 37 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that the function $y=f\left(x\right)$ is increasing on the interval $(-1,5)$ 

(a) Over what interval is the graph of $y=f(x+2)$ increasing?

(b) Over what interval is the graph of $y=f(x-5)$ increasing?

(c) What can be said about the graph of $y=-f\left(x\right)$?

(d) What can be said about the graph of $y=f(-x)$? 

Step-by-Step Solution

Answer

$y=f(x+2)$ is increasing in the interval $(-3,3)$

$y=f(x-5)$ is increasing in the interval $(4,10)$

$y=-f\left(x\right)$ is decreasing in the interval $(-1,5)$

$y=f(-x)$ is decreasing in the interval $(-5,1)$

Part (a) Step 1. Find the interval

Interval on which $y=f(x+2)$ can be found as:

$$\begin{gathered} x+2=-1 \\ x=-3 \end{gathered}$$

Interval is $(-3,3)$

and,

$$\begin{gathered} x+2=5 \\ x=3 \end{gathered}$$

Part (b) Step 1. Find the interval

The graph of $y=f(x-5)$ is shifted toward the right, hence the interval can be found as:

$$\begin{gathered} x-5=-1 \\ x=4 \end{gathered}$$

and

$$\begin{gathered} x-5=5 \\ x=10 \end{gathered}$$

Interval is $(4,10)$

Part (c) Step 1. Find the interval

The graph of $y=-f\left(x\right)$ is giving negative sign to the output of $y=f\left(x\right)$.

Therefore where $y=f\left(x\right)$ is increasing, $y=-f\left(x\right)$ is decreasing.

Therefore,

The interval is $(-1,5)$

Part (d) Step 1.

The graph of $y=f(-x)$ always gives negative sign to the input of $y=f\left(x\right)$.

Hence the interval is $(-5,1)$