Q. 38 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that the function $y=f\left(x\right)$ is decreasing on the interval $\left(-2, 7\right)$.

1. Over what interval is the graph of $y=f(x+2)$ decreasing?
2. Over what interval is the graph of $y=f(x-5)$ decreasing?
3. What can be said about the graph of $y=-f\left(x\right)$ ?
4. What can be said about the graph of $y=f(-x)$ ?

Step-by-Step Solution

Answer

1. The function $f(x+2)$ is decreasing in the interval $(-4,5)$.
2. The function $f(x-5)$ is decreasing in the interval $\left(3,12\right)$.
3. The function $y=-f\left(x\right)$ is increasing in the interval $\left(-2,7\right)$.
4. The function $f(-x)$ is increasing in the interval $\left(-7,2\right)$
   

Step 1. Given Information

We have given that the function $y=f\left(x\right)$ 

is decreasing on the interval $(-2,7)$

Part (a) Step 1. Finding the interval

The graph of function $y=f(x+2)$ is shifted  $2$ units towards left, so the interval on which it is decreasing is given by,

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

And

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

So, the decreasing interval is $(-4,5)$.

Part (b) Step 1. Finding the interval

The graph of $y=f(x-5)$ is shifted $5$ units towards right, so the interval on which it is decreasing is given by,

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

Similarly,

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

So, the decreasing interval is $(3,12)$.

Part(c) Step 1. About - f ( x )

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

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

Part(d) Step 1. About y = f ( - x )

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

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