Question

Show that the functions \(f\) and \(g\) are inverses of each other by graphing them in the same viewing window. $$ f(x)=e^{x-1}, g(x)=1+\ln x $$

Short answer

The functions \( f(x)=e^{x-1} \) and \( g(x)=1+\ln x \) are inverses of each other. This has been proven analytically by showing that \( f(g(x)) = x \) and \( g(f(x)) = x \), and then visually by graphing the two functions, showing they reflect each other along the line \( y = x \).

Step-by-step solution

Compute \( f(g(x)) \)

For \( f(g(x)) \), replace every x in f(x) with g(x). Therefore, you get \( f(g(x)) = e^{g(x)-1} = e^{(1+\ln x)-1} = e^{\ln x} = x \).

Compute \( g(f(x)) \)

For \( g(f(x)) \), replace every x in g(x) with f(x). Therefore, you get \( g(f(x)) = 1 + \ln f(x) = 1 + \ln e^{x-1} = 1 +(x-1) = x \).

Graph both functions

To graph the functions, select an appropriate viewing window and plot the points for each function. Since both \( g(f(x)) \) and \( f(g(x)) \) equal x, as a result when you graph \( f(x) = e^{x-1} \) and \( g(x) = 1+\ln x \) on the same viewing window, the graph of g will be a reflection of f across the line \( y = x \), confirming that they are inverses.