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.