If the functions f and g defined from the set of real number R to R such that `f(x) = e^(x)` and g(x) = 3x - 2, then find functions fog and gof.

Explanation

To find the composite functions \( f \circ g \) and \( g \circ f \) for the given functions \( f(x) = e^x \) and \( g(x) = 3x - 2 \), we will follow these steps:

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

The notation \( f \circ g \) means we need to apply the function \( f \) to the output of the function \( g \). This can be expressed as:

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

### Step 2: Substitute \( g(x) \) into \( f(x) \)

We know that:

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

Now, we substitute \( g(x) \) into \( f(x) \):

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

### Step 3: Apply the function \( f \)

Since \( f(x) = e^x \), we can write:

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

Thus, we have:

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

### Step 4: Find \( g \circ f \)

Now we need to find \( g \circ f \), which means we apply the function \( g \) to the output of the function \( f \):

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

### Step 5: Substitute \( f(x) \) into \( g(x) \)

We know that:

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

Now, we substitute \( f(x) \) into \( g(x) \):

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

### Step 6: Apply the function \( g \)

Since \( g(x) = 3x - 2 \), we can write:

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

Thus, we have:

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

### Final Results

The composite functions are:

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

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