Chapter 6.4, Problem 21Q

To determine

To write: The rule to draw the graph of the function $g$ if $f(x)={\log }_{3}x$ .

Answer to Problem 21Q

The rule for the function $g$ is $g\left(x\right)={\log }_3\left(x-2\right)+1$ .

Explanation of Solution

Given information:

The graph of $g$ is a transformation of the graph of $f$ and the function is $f\left(x\right)={\log }_{3}x$ .

  

Calculation:

The graph of the function $f\left(x\right)={\log }_{3}x$ is drawn below.

  

From the graph above graph, it is clear that the graph of the function $g\left(x\right)$ is formed by a horizontal translation 2 units to the right and vertical translation 1 unit upward form the function $f\left(x\right)$ .

Let the translation of the function $f\left(x\right)$ is $h\left(x\right)$ . Write the function $h\left(x\right)$ .

  $h\left(x\right)=f\left(x-2\right)$

Substitute the value of $f\left(x\right)$ in the above function.

  $h\left(x\right)={\log }_3\left(x-2\right)$

Write a function $g\left(x\right)$ that represents the vertical translation of $h\left(x\right)$ .

  $g\left(x\right)=h\left(x\right)+1$

Substitute the value of $h\left(x\right)$ in the above equation.

  $g\left(x\right)={\log }_3\left(x-2\right)+1$

Therefore, the rule for the function $g$ is $g\left(x\right)={\log }_3\left(x-2\right)+1$ .