If the function f and g are defined from the set of real numbers R to R such that `f(x)=e^x, g(x) = 3x - 2` then find functions fog and gof. Also find the domains of functions `(fog)^(-1)` and `(g o f)^(-1)`.

Answer

To solve the problem, we need to find the functions \( f \circ g \) and \( g \circ f \) given the functions \( f(x) = e^x \) and \( g(x) = 3x - 2 \). After that, we will find the domains of the inverse functions \( (f \circ g)^{-1} \) and \( (g \circ f)^{-1} \).

### Step 1: Find \( f \circ g \)

The composition \( f \circ g \) means we will substitute \( g(x) \) into \( f(x) \).

\[ f \circ g(x) = f(g(x)) = f(3x - 2) \]

Now, substituting \( g(x) \) into \( f(x) \):

\[ f(3x - 2) = e^{3x - 2} \]

So,

\[ f \circ g(x) = e^{3x - 2} \]

### Step 2: Find \( g \circ f \)

Next, we find \( g \circ f \) by substituting \( f(x) \) into \( g(x) \).

\[ g \circ f(x) = g(f(x)) = g(e^x) \]

Now, substituting \( f(x) \) into \( g(x) \):

\[ g(e^x) = 3(e^x) - 2 = 3e^x - 2 \]

So,

\[ g \circ f(x) = 3e^x - 2 \]

### Step 3: Find the Inverse of \( f \circ g \)

To find \( (f \circ g)^{-1} \), we start with:

\[ y = e^{3x - 2} \]

Now, we solve for \( x \):

1. Take the natural logarithm of both sides:
\[ \ln(y) = 3x - 2 \]

2. Rearranging gives:
\[ 3x = \ln(y) + 2 \]

3. Finally, divide by 3:
\[ x = \frac{\ln(y) + 2}{3} \]

Thus, the inverse function is:

\[ (f \circ g)^{-1}(y) = \frac{\ln(y) + 2}{3} \]

### Step 4: Find the Domain of \( (f \circ g)^{-1} \)

The domain of \( (f \circ g)^{-1} \) is determined by the range of \( f \circ g(x) = e^{3x - 2} \). Since the exponential function is always positive:

\[ y > 0 \]

Therefore, the domain of \( (f \circ g)^{-1} \) is:

\[ (0, \infty) \]

### Step 5: Find the Inverse of \( g \circ f \)

To find \( (g \circ f)^{-1} \), we start with:

\[ y = 3e^x - 2 \]

Now, we solve for \( x \):

1. Rearranging gives:
\[ 3e^x = y + 2 \]

2. Dividing by 3:
\[ e^x = \frac{y + 2}{3} \]

3. Taking the natural logarithm of both sides:
\[ x = \ln\left(\frac{y + 2}{3}\right) \]

Thus, the inverse function is:

\[ (g \circ f)^{-1}(y) = \ln\left(\frac{y + 2}{3}\right) \]

### Step 6: Find the Domain of \( (g \circ f)^{-1} \)

The domain of \( (g \circ f)^{-1} \) is determined by the range of \( g \circ f(x) = 3e^x - 2 \). Since \( e^x \) is always positive, the minimum value of \( g \circ f(x) \) occurs when \( e^x \) approaches 0:

\[ 3(0) - 2 = -2 \]

Thus, \( y \) must satisfy:

\[ y > -2 \]

Therefore, the domain of \( (g \circ f)^{-1} \) is:

\[ (-2, \infty) \]

### Summary of Results:

1. \( f \circ g(x) = e^{3x - 2} \)
2. \( g \circ f(x) = 3e^x - 2 \)
3. Domain of \( (f \circ g)^{-1} \): \( (0, \infty) \)
4. Domain of \( (g \circ f)^{-1} \): \( (-2, \infty) \)