https://file.aiquickdraw.com/mathbot/file/17262637507254m4p37t1.jpg

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!

أي الدوال التالية زوجي؟ (Which of the following functions is even?)

Determine which function is even by using the condition f(-x) = f(x). This problem covers key algebraic concepts such as even and odd functions, symmetry, and function analysis. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation