Chapter 4.3, Problem 59E

To determine

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

Answer to Problem 59E

The required graph is shown in Figure 3, 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 $x=1$ .

Explanation of Solution

Given:

The given function is $y=1-{\log }_{10}x$ .

Calculation:

Consider the parent function for given function is,

  $y={\log }_{10}x$

The graph for the given parent function $y={\log }_{10}x$ is,

  $y=1-{\log }_{10}x$

The graph for the parent function is shown in Figure 1

Figure 1

Reflect the parent function $y={\log }_{10}x$ across the x axis and the plot for it is shown in Figure 2

Figure 2

Thus, the function for the above graph is $g\left(x\right)=-{\log }_{10}x$ . To obtain $g\left(x\right)=1-{\log }_{10}x$ shift the graph $g\left(x\right)=-{\log }_{10}x$ one unit in the up direction.

The required graph is shown in Figure 3

Figure 3

From the above graph, the asymptote of the function $g\left(x\right)=1-{\log }_{10}x$ is,

  $x=0$

The domain of the function is,

  $\left(0,\infty \right)$

The range of the given function $g\left(x\right)=1-{\log }_{10}x$ is,

  $\left(-\infty ,\infty \right)$