The problem asks to determine whether the function $f(x) = 3x^2 + 52x - 3$ is odd, even, both, or neither. Here's how we solve it:

Step 1: Define what it means for a function to be even or odd:

• A function is even if $f(-x) = f(x)$ for all $x$.
• A function is odd if $f(-x) = -f(x)$ for all $x$.

Step 2: Calculate $f(-x)$

For the given function $f(x) = 3x^2 + 52x - 3$, substitute $-x$ into the function: $f(-x) = 3(-x)^2 + 52(-x) - 3 = 3x^2 - 52x - 3$

Step 3: Compare $f(-x)$ with $f(x)$

• $f(x) = 3x^2 + 52x - 3$
• $f(-x) = 3x^2 - 52x - 3$

Clearly, $f(-x) \neq f(x)$, so the function is not even.

Step 4: Check if $f(-x) = -f(x)$

Now, check if $f(-x)$ is the negative of $f(x)$: $-f(x) = -(3x^2 + 52x - 3) = -3x^2 - 52x + 3$ Clearly, $f(-x) \neq -f(x)$, so the function is not odd.

Conclusion:

The function $f(x) = 3x^2 + 52x - 3$ is neither even nor odd.

Would you like more details on this? Here are some related questions:

1. What are the steps to identify an even or odd function for other types of functions?
2. Can a function be both odd and even?
3. How can symmetry of graphs help in identifying even and odd functions?
4. What types of functions are typically even or odd?
5. How does the behavior of polynomial degrees affect whether a function is even or odd?

Tip: Remember, even functions are symmetric about the y-axis, while odd functions are symmetric about the origin!