Which of the following is not an even function?

Options

A. `f(x)=e^(x) +e^(-x)`
B. `f(x) =e^(x) -e^(-x)`
C. `f(x)= e^(2x) +e^(-2x)`
D. None of these

Answer

To determine which of the given functions is not an even function, we need to analyze each function. An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \).

Let's go through the functions step by step:

### Step 1: Analyze the first function
**Function:** \( f(x) = e^x + e^{-x} \)

**Step 1.1:** Calculate \( f(-x) \):
\[ f(-x) = e^{-x} + e^{x} \]

**Step 1.2:** Simplify \( f(-x) \):
\[ f(-x) = e^x + e^{-x} = f(x) \]

**Conclusion for the first function:** Since \( f(-x) = f(x) \), this function is an even function.

### Step 2: Analyze the second function
**Function:** \( f(x) = e^x - e^{-x} \)

**Step 2.1:** Calculate \( f(-x) \):
\[ f(-x) = e^{-x} - e^{x} \]

**Step 2.2:** Simplify \( f(-x) \):
\[ f(-x) = - (e^x - e^{-x}) = -f(x) \]

**Conclusion for the second function:** Since \( f(-x) = -f(x) \), this function is not an even function.

### Final Conclusion
The function that is not an even function is **option B**: \( f(x) = e^x - e^{-x} \).

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