Chapter 3.2, Problem 86E

To determine

Whether the given statement “the graph of $f(x)=\ln (-x)$ is a reflection of the graph $h(x)=e^{-x}$ in the line $y=-x$ ” is true or false.

Answer to Problem 86E

The given statement is true.

Explanation of Solution

Given information:

The graph of $f(x)=\ln (-x)$ is a reflection of the graph $h(x)=e^{-x}$ in the line $y=-x$ .

Since we know that the function $f(x)=\ln (x)$ is inverse function of $h(x)=e^x$

Thus, the graph of $f(x)=\ln (x)$ is a reflection of the graph $h(x)=e^x$ in the line $y=x$ .

Now replace $x$ by $-x$ , we get

The, graph of $f(x)=\ln (-x)$ is a reflection of the graph $h(x)=e^{-x}$ in the line $y=-x$ .

This can verify by the following graph of the functions $f(x)=\ln (-x),h(x)=e^{-x}\text{and}y=-x$

Graph:

  

Interpretation: from above graph, we observe that the graph of $f(x)=\ln (-x)$ is a reflection of the graph $h(x)=e^{-x}$ in the line $y=-x$

Therefore, the given statement is true.