Chapter 4.3, Problem 57E

To determine

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

Answer to Problem 57E

The required graph is shown in Figure 1, the domain of the function is $\left(0,\infty \right)$ and its range is $\left(-\infty ,-\infty \right)$ . The vertical asymptotes of the graph is $x=0$ .

Explanation of Solution

Given:

The given function is $y=2+{\log }_{3}x$ .

Calculation:

Consider the given function is,

  $g\left(x\right)=2+{\log }_{3}x$

Consider the parent function is,

  $f\left(x\right)={\log }_{3}x$

The graph for $g\left(x\right)=2+{\log }_{3}x$ is obtained by shifting the graph $f\left(x\right)={\log }_{3}x$ two units upward

The domain of the function is the state values for which the function is defined as the value of the logarithm function is undefined for the negative values. Thus, the domain of $f\left(x\right)={\log }_{3}x$ is,

  $\left(0,\infty \right)$

Also, the graph for the function $g\left(x\right)=2+{\log }_{3}x$ is shifted upward, thus, it does not affect the domain.

  $\left(0,\infty \right)$

The range of the function is the real line that is the range of $g\left(x\right)=2+{\log }_{3}x$ is,

  $\left(-\infty ,\infty \right)$

The graph for the given function is shown in Figure 1

Figure 1

From the above graph the vertical asymptote of the graph is,

  $x=0$