Which of the following is an even functions?

Options

A. A)`|x^(2)|-5x`
B. B)`x^(4)+x^(5)`
C. C)`e^(2x) +e^(-2x)`
D. D)`|x|^(2)//x`

Answer

To determine which of the given functions is an even function, we need to apply the definition of even functions. An even function \( f(x) \) satisfies the condition:

\[ f(x) = f(-x) \]

Let's analyze the options provided, focusing on option C, which is given as:

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

### Step 1: Substitute \(-x\) into the function
To check if \( f(x) \) is even, we need to compute \( f(-x) \):

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

### Step 2: Simplify the expression
Now, simplify the expression we obtained:

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

### Step 3: Rearrange the terms
Notice that we can rearrange the terms:

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

### Step 4: Compare \( f(-x) \) with \( f(x) \)
Now, we compare \( f(-x) \) with \( f(x) \):

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

Since \( f(-x) = f(x) \), we conclude that:

\[ f(x) \text{ is an even function.} \]

### Conclusion
Thus, the function \( e^{2x} + e^{-2x} \) (option C) is an even function.

---