The question in the image asks: "Which of the following functions is even?" The functions provided are:

• $f(x) = x^3$ (B)
• $f(x) = \frac{1}{x}$ (D)
• $f(x) = x^2 + |x|$ (A)
• $f(x) = x^2 + x$ (C)

To determine if a function is even:

A function $f(x)$ is even if: $f(-x) = f(x) \text{ for all } x.$

Now, let's test each function to see if it satisfies this condition:

1. $f(x) = x^3$:

   • $f(-x) = (-x)^3 = -x^3$, which is not equal to $x^3$. Therefore, this function is not even.

2. $f(x) = \frac{1}{x}$:

   • $f(-x) = \frac{1}{-x} = -\frac{1}{x}$, which is not equal to $\frac{1}{x}$. Therefore, this function is not even.

3. $f(x) = x^2 + |x|$:

   • $f(-x) = (-x)^2 + |-x| = x^2 + |x|$. This is the same as $f(x)$. Therefore, this function is even.

4. $f(x) = x^2 + x$:

   • $f(-x) = (-x)^2 + (-x) = x^2 - x$, which is not equal to $x^2 + x$. Therefore, this function is not even.

Conclusion:

The correct answer is $\mathbf{A}$, $f(x) = x^2 + |x|$, which is an even function.

Would you like further clarification or additional details on how to test for even and odd functions?

Here are some related questions:

1. How do you test if a function is odd?
2. What are the general properties of even and odd functions?
3. Can a function be both even and odd?
4. What does the graph of an even function look like?
5. How are absolute values involved in even functions?

Tip: Even functions are symmetric about the y-axis. This can help you quickly recognize them visually!