If `f(x)=cosx and g(x)=2x+1`, which of the following are even functions?
I. `f(x)*g(x)`
II `f(g(x))`
III. `g(f(x))`

Options

A. only I
B. only II
C. only III
D. Only I and II

Explanation

To determine which of the given functions are even functions, we need to recall the definition of an even function. A function \( f(x) \) is considered even if \( f(-x) = f(x) \) for all \( x \) in the domain of the function.

Given:
- \( f(x) = \cos x \)
- \( g(x) = 2x + 1 \)

We will analyze each of the three functions one by one.

### Step 1: Check \( f(x) \cdot g(x) \)

1. **Calculate \( f(-x) \cdot g(-x) \)**:
- \( f(-x) = \cos(-x) = \cos x \) (since cosine is an even function)
- \( g(-x) = 2(-x) + 1 = -2x + 1 \)

Therefore,
\[ f(-x) \cdot g(-x) = \cos x \cdot (-2x + 1) \]

2. **Calculate \( f(x) \cdot g(x) \)**:
- \( g(x) = 2x + 1 \)

Therefore,
\[ f(x) \cdot g(x) = \cos x \cdot (2x + 1) \]

3. **Compare \( f(-x) \cdot g(-x) \) and \( f(x) \cdot g(x) \)**:
\[ f(-x) \cdot g(-x) = \cos x \cdot (-2x + 1) \quad \text{and} \quad f(x) \cdot g(x) = \cos x \cdot (2x + 1) \]
Since \( \cos x \cdot (-2x + 1) \neq \cos x \cdot (2x + 1) \), \( f(x) \cdot g(x) \) is **not an even function**.

### Step 2: Check \( f(g(x)) \)

1. **Calculate \( f(g(-x)) \)**:
- \( g(-x) = -2x + 1 \)
- Therefore, \( f(g(-x)) = f(-2x + 1) = \cos(-2x + 1) \)

2. **Calculate \( f(g(x)) \)**:
- \( g(x) = 2x + 1 \)
- Therefore, \( f(g(x)) = f(2x + 1) = \cos(2x + 1) \)

3. **Compare \( f(g(-x)) \) and \( f(g(x)) \)**:
\[ f(g(-x)) = \cos(-2x + 1) \quad \text{and} \quad f(g(x)) = \cos(2x + 1) \]
Since \( \cos(-2x + 1) \neq \cos(2x + 1) \), \( f(g(x)) \) is **not an even function**.

### Step 3: Check \( g(f(x)) \)

1. **Calculate \( g(f(-x)) \)**:
- \( f(-x) = \cos(-x) = \cos x \)
- Therefore, \( g(f(-x)) = g(\cos x) = 2\cos x + 1 \)

2. **Calculate \( g(f(x)) \)**:
- Therefore, \( g(f(x)) = g(\cos x) = 2\cos x + 1 \)

3. **Compare \( g(f(-x)) \) and \( g(f(x)) \)**:
\[ g(f(-x)) = 2\cos x + 1 \quad \text{and} \quad g(f(x)) = 2\cos x + 1 \]
Since \( g(f(-x)) = g(f(x)) \), \( g(f(x)) \) is an **even function**.

### Conclusion:
- The only function that is an even function among the given options is **III. \( g(f(x)) \)**.

### Final Answer:
The correct option is **III only**.

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