Chapter 4.3, Problem 53E

To determine

To find: The graph for the function and the find he domain, range, and asymptote.

Answer to Problem 53E

The required graph is shown in Figure 2, the domain of the function is

  $\left\{x|x-4>0\right\}\text{or{x|x>4}or{4,}\infty \text{}}$ and its range is $ℝ\text{or}\left(-\infty ,\infty \right)$ . The vertical asymptote of the function is $x=4$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)={\log }_2\left(x-4\right)$ .

Calculation:

Consider the parent function is,

  $f\left(x\right)={\log }_2\left(x\right)$

The graph for the equation $f\left(x\right)={\log }_2\left(x\right)$ is shown in Figure 1

Figure 1

The graph for the function ${\log }_2\left(x-4\right)$ is obtained from the parent graph by shifting the graph of ${\log }_{2}x$ to the right by 4 units.

Substitute $4$ for $x$ in the equation ${\log }_2\left(x-4\right)$ .

  $\begin{array}{c}f\left(4\right)={\log }_2\left(4-4\right)\\ ={\log }_{2}0\\ =\infty \end{array}$

Thus, the value $x=4$ is the vertical asymptote of $f\left(x\right)$ . The value of the function ${\log }_{2}x$ is defined for the only region $x>0$ . The function is not defined for the $x\le 4$ .

Then the domain of the function $f\left(x\right)={\log }_2\left(x-4\right)$ is,

  $\left\{x|x-4>0\right\}\text{or{x|x>4}or{4,}\infty \text{}}$

The values of the domain of $f\left(x\right)$ has the values for the whole real set. Thus, the range of $f\left(x\right)$ is,

  $ℝ\text{or}\left(-\infty ,\infty \right)$

The graph for the function $f\left(x\right)={\log }_2\left(x-4\right)$ is shown in Figure 2

Figure 2