Chapter 6.4, Problem 40E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)={\log }_{5}x$ ; reflection in the $x-$ axis, followed by a translation $9$ units left.

Answer to Problem 40E

The rule for $g$ that represents the indicated transformation of the graph $f$ is:

  $g\left(x\right)=-{\log }_5\left(x+9\right)$

Explanation of Solution

Given information:

The given function is $f\left(x\right)={\log }_{5}x$ .

Calculation:

Consider the given function

  $f\left(x\right)={\log }_{5}x$ .

First, reflect the function across the $x-$ axis.

For reflecting the function across the $x-$ axis, change the sign of the function.

Therefore, after reflection in the $x-$ axis, the function becomes $f\left(x\right)=-{\log }_{5}x$ .

After that, translate the graph left by the multiplication of the function by $9$ unit.

For the translating left, addthe given unit to the $x$ value of the function $f$ .

Therefore after the translation, the function becomes $f\left(x\right)=-{\log }_5\left(x+9\right)$ .

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=-{\log }_5\left(x+9\right)$ .