Question

Use the given function $f\left(x\right)=3e^{x-2}$to:

(a) Find the domain of f.

(b) Graph f.

(c) From the graph, determine the range and any asymptotes of f.

(d) Find $f^{-1}$, the inverse of f.

(e) Find the domain and the range of $f^{-1}$.

(f) Graph $f^{-1}$.

Short answer

Part (a) Domain of f:$(-\infty ,\infty )$

Part (b) Graph of f:

Part (c) Range of f: $(0,\infty )$

Horizontal asymptote:$y=0$

Part (d)$f^{-1}\left(x\right)=2+\ln \left(\frac{x}{3}\right)$

Part (e) Domain of $f^{-1}:(0,\infty )$

Range of$f^{-1}:(-\infty ,\infty )$

Part (f) Graph of$f^{-1}:$

Step-by-step solution

Part (a) Step 1. Given information 

A function,$f\left(x\right)=3e^{x-2}$

Part (a) Step 2. Domain of the function

$x$can take any real value. Therefore, domain is: $(-\infty ,\infty )$.

Part (b) Step 1. Graph of the function

Graph of the function is:

Part (c) Step 1. Range and asymptote

From the graph, for the given domain, range of f : $(0,\infty )$

Horizontal asymptote: $y=0$

Part (d) Step 1. Inverse of function

In the function, replace x by $f^{-1}\left(x\right)$and $f\left(x\right)$by x and rearrange the variables to get inverse of the function.

$x=3e^{\left(f^{-1}\right(x)-2)}$

$\frac{x}{3}=e^{f^{-1}\left(x\right)-2}$

taking log to the base e on both sides

$\ln \left(\frac{x}{3}\right)=\left[f^{-1}\left(x\right)-2\right]\ln e$

$f^{-1}\left(x\right)=2+\ln \left(\frac{x}{3}\right)$

Part (e) Step 1. Domain and Range of inverse function

$f^{-1}\left(x\right)=2+\ln \left(\frac{x}{3}\right)$

$\frac{x}{3}>0\Rightarrow x>0$

Therefore, domain of $f^{-1}:(0,\infty )$

Range of$f^{-1}:(-\infty ,\infty )$

Part (f) Step 1. Graph of inverse function

Graph of inverse function is: