#### To determine

**To find:** Find (a) the domain of the function f(b) f-1 and its domain.

#### Answer

The domain of the function is ln3,∞. The inverse function is f-1(x)=ln(ex+3) and its domain is all real number.

#### Explanation

**Calculation: fx=ln(ex-3)**

**(a)**

The given function *f* is defined whenever ex-3>0. i.e. ex>3⇔x>ln3.

Therefore, domain of the function *f* is ln3,∞.

**(b)**

To find the inverse function f-1 and its domain, we first write the function as y=ln(ex-3). Then we solve the equation for x.

Applying the exponential function on both sides, we have

ey=eln(ex-3)

ey=(ex-3)

ex=ey+3

Taking log both sides

x=ln(ey+3)

We interchange *x* and *y*:

y=ln(ex+3)

Therefore, the inverse function is f-1(x)=ln(ex+3).

The inverse function is defined whenever (ex+3)>0. Since ex>0. Therefore, the domain of f-1 is set of all real number.

**Final statement:**

The domain of the function is ln3,∞. The inverse function is

f-1(x)=ln(ex+3) and its domain is all real number.