Determine if the function is even odd or neither.
f(x) = 5x^2 + 3

To determine if a function is **even**, **odd**, or **neither**, we use the following definitions:

- A function is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function is **odd** if \( f(-x) = -f(x) \) for all \( x \).

Given the function \( f(x) = 5x^2 + 3 \), let's test it.

### Step 1: Check if the function is even
To check for evenness, substitute \( -x \) into the function:

\[
f(-x) = 5(-x)^2 + 3 = 5x^2 + 3
\]

Since \( f(-x) = f(x) \), the function is **even**.

### Step 2: Check if the function is odd
For oddness, we need to check if \( f(-x) = -f(x) \):

\[
f(-x) = 5x^2 + 3
\]
\[
-f(x) = -(5x^2 + 3) = -5x^2 - 3
\]

Clearly, \( f(-x) \neq -f(x) \), so the function is **not odd**.

### Conclusion:
The function \( f(x) = 5x^2 + 3 \) is **even**.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What is the graphical interpretation of an even function?
2. Can a function be both even and odd? Why or why not?
3. What would happen if the function was \( f(x) = 5x^2 + 3x \)?
4. How do transformations (like translations) affect the even or odd nature of a function?
5. What are some real-world applications of even and odd functions?

**Tip:** An even function is always symmetric about the y-axis, while an odd function is symmetric about the origin.

Determine if the function is even, odd, or neither. f(x) = 5x^2 + 3

Explore how to determine whether the function f(x) = 5x^2 + 3 is even, odd, or neither. The step-by-step solution covers key algebraic concepts related to functions, including definitions and examples of even and odd functions. Suitable for students in Grades 9-12.

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