#### 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 ln32,∞. The inverse function is

f-1(x)=ln(3-x2)2 and its domain is [0,3).

#### Explanation

**Calculation:**

fx=3-e2x

**(a)**

The given function *f* is defined whenever 3-e2x>0. i.e. 3>e2x. Taking the natural logarithm on

both sides of 3>e2x, we have

ln32>x

Therefore, domain of the function *f* is ln32,∞.

**(b)**

To find the inverse function f-1 and its domain, we first write the function as

y=3-e2x. Then we solve the equation for x.

y=3-e2x

y2=3-e2x or e2x=3-y2 (1)

Taking the natural log both sides of the equation (1), we have

ln e2x=ln(3-y2)

2x=ln(3-y2)

x=ln(3-y2)2

Finally, we interchange x and y:

y=ln(3-x2)2

Therefore, the inverse function is

f-1(x)=ln(3-x2)2

The domain of f-1 is defined when 3-x2>0 or 3>x2**or 3>x and x>-3**.i.e.(-3,3).

So, the domain of the function is -3,3 but the domain of the inverse function is the range of the function *f(x)* and *f(x)* is the square root function whose range can not be negative. So, domain of the inverse function is [0,3).

**Final statement:** The domain of the function is ln32,∞. The inverse function is

f-1(x)=ln(3-x2)2 and its domain is [0,3).