#### 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 e-2,∞. The inverse function is

f-1(x)=e(ex-2) and its domain is all real number.

#### Explanation

**Calculation: fx=ln(2+lnx)**

**(a)**

The given function *f* is defined whenever 2+lnx>0. i.e. lnx>-2.

as lnx is defined for x>0. Applying the exponential function on both sides, we have

x>e-2.

Therefore, domain of the function *f* is e-2,∞.

**(b)**

To find the inverse function f-1 and its domain, we first write the function as y=ln(2+lnx). Then we solve the equation for x.

Applying the exponential function on both sides, we have

ey=eln(2+lnx)

ey=(2+lnx)

lnx=ey-2

Again, applying the exponential function

x=eey-2

We interchange x and y:

y=eex-2. Therefore, the inverse function is

f-1(x)=e(ex-2)

The domain of f-1(x)=e(ex-2) is set of all real number since exponential is defined for all real number.

**Final statement:**

The domain of the function is e-2,∞. The inverse function is

f-1(x)=e(ex-2) and its domain is all real number.