Function `f: R to R and g : R to R ` are defined as `f(x)=sin x and g(x) =e^(x)` .
Find (gof)(x) and (fog)(x).

Answer

To solve the problem, we need to find the compositions of the functions \( g \) and \( f \), specifically \( (g \circ f)(x) \) and \( (f \circ g)(x) \).

### Step 1: Identify the functions
We are given:
- \( f(x) = \sin x \)
- \( g(x) = e^x \)

### Step 2: Find \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means we apply \( f \) first and then apply \( g \) to the result of \( f \).

1. Start with \( f(x) \):
\[ f(x) = \sin x \]

2. Now, substitute \( f(x) \) into \( g \):
\[ g(f(x)) = g(\sin x) \]

3. Using the definition of \( g \):
\[ g(\sin x) = e^{\sin x} \]

Thus, we have:
\[ (g \circ f)(x) = e^{\sin x} \]

### Step 3: Find \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we apply \( g \) first and then apply \( f \) to the result of \( g \).

1. Start with \( g(x) \):
\[ g(x) = e^x \]

2. Now, substitute \( g(x) \) into \( f \):
\[ f(g(x)) = f(e^x) \]

3. Using the definition of \( f \):
\[ f(e^x) = \sin(e^x) \]

Thus, we have:
\[ (f \circ g)(x) = \sin(e^x) \]

### Final Answers
- \( (g \circ f)(x) = e^{\sin x} \)
- \( (f \circ g)(x) = \sin(e^x) \)