Chapter 3, Problem 36RE

Solution

To explain : The transformation of the graph of $y={\log }_{2}x$ into the graph of $g\left(x\right)={\log }_2\left(4-x\right)$ and the graph.

The graph of $g\left(x\right)={\log }_2\left(4-x\right)$ is obtained from the graph of $y={\log }_{2}x$ by applying in order, a reflection across the $y-$ axis followed by translation $4$ units to the right.

Given information :

The function: $g\left(x\right)={\log }_2\left(4-x\right)$ .

Formula used:

The translation of a function $y=f\left(x\right)$ into the form $y=f\left(x\pm k\right)$ , represents the horizontal translation. If $k>0$ , then the function moves to the left side by $\text{'}k\text{'}$ units and if $k<0$ , then the function moves to the right by $\text{'}k\text{'}$ units.

  $y=-f\left(x\right)$ is the reflection of $y=f\left(x\right)$ with respect to the $x-$ axis.

  $y=f\left(-x\right)$ is the reflection of $y=f\left(x\right)$ with respect to the $y-$ axis.

Explanation :

Consider the logarithmic equation,

  $\begin{array}{c}g\left(x\right)={\log }_2\left(4-x\right)\\ ={\log }_2\left[-\left(x-4\right)\right]\end{array}$

So, we obtain the graph of $g\left(x\right)={\log }_2\left(4-x\right)$ from the graph of $y={\log }_{2}x$ by applying in order, a reflection across the $y-$ axis followed by translation $4$ units to the right.

Sketch the graph.