Chapter 6.4, Problem 41E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph.

Answer to Problem 41E

The rule for $g$ that represents the indicated transformation of the graph $f$ is $f\left(x\right)={\log }_{\frac{1}{2}}\left(-x+3\right)+2$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)={\log }_{\frac{1}{2}}x$ .

Calculation:

Consider the given function $f\left(x\right)={\log }_{\frac{1}{2}}x$ .

First,translate the function $3$ units left. Add 3 on the $x-$ coordinate to translate the function $3$ units left.

After the translation the function becomes $f\left(x\right)={\log }_{\frac{1}{2}}\left(x+3\right)$ .

Now, the obtained function has to translate 2 units up. Add $2$ on the $y-$ coordinate of the obtained function to translate the graph of the function2 units up.

After the translation the function becomes $f\left(x\right)={\log }_{\frac{1}{2}}\left(x+3\right)+2$ .

Now, the graph has to be reflected through the $y-$ axis. Change the sign of the $x-$ coordinate to reflect the graph of the function over the $y-$ axis.

After the reflection the graph of the function becomes $f\left(x\right)={\log }_{\frac{1}{2}}\left(-x+3\right)+2$ .

Hence the rule for $g$ that represents the indicated transformation of the graph $f$ is $f\left(x\right)={\log }_{\frac{1}{2}}\left(-x+3\right)+2$ .