If f is an even function and g is an odd function then the function fog is

Options

A. an even function
B. an odd function
C. neither even nor odd
D. a periodic funtion

Answer

To determine whether the composition of functions \( f \) and \( g \) (denoted as \( f \circ g \)) is even, odd, or neither, we start by using the definitions of even and odd functions.

### Step 1: Understand the properties of the functions
- An **even function** \( f \) satisfies the property:
\[ f(-x) = f(x) \quad \text{for all } x \]
- An **odd function** \( g \) satisfies the property:
\[ g(-x) = -g(x) \quad \text{for all } x \]

### Step 2: Find \( f(g(-x)) \)
To analyze \( f \circ g \), we need to evaluate \( f(g(-x)) \):
\[ g(-x) = -g(x) \quad \text{(since \( g \) is odd)} \]
Thus, we can substitute this into \( f \):
\[ f(g(-x)) = f(-g(x)) \]

### Step 3: Use the property of the even function \( f \)
Since \( f \) is an even function, we have:
\[ f(-g(x)) = f(g(x)) \quad \text{(since \( f \) is even)} \]

### Step 4: Conclusion
Putting it all together, we find:
\[ f(g(-x)) = f(g(x)) \]
This shows that \( f \circ g \) satisfies the property of an even function.

### Final Result
Thus, the function \( f \circ g \) is an **even function**.

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