Determine whether the following function is even, odd, or neither. f ( x )

To determine whether the function $f(x) = \frac{1}{4}x^3 + x^2 - x - 3$ is even, odd, or neither, we follow these steps:

1. Check for an even function: A function is even if $f(-x) = f(x)$ for all $x$.

2. Check for an odd function: A function is odd if $f(-x) = -f(x)$ for all $x$.

Step 1: Compute $f(-x)$

Substitute $-x$ into the function $f(x)$:

$f(-x) = \frac{1}{4}(-x)^3 + (-x)^2 - (-x) - 3$

Simplifying each term: $f(-x) = \frac{1}{4}(-x^3) + x^2 + x - 3$ $f(-x) = -\frac{1}{4}x^3 + x^2 + x - 3$

Step 2: Compare $f(-x)$ with $f(x)$

We have:

• $f(x) = \frac{1}{4}x^3 + x^2 - x - 3$
• $f(-x) = -\frac{1}{4}x^3 + x^2 + x - 3$

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

Step 3: Check if the function is odd

Now, let's check if $f(-x) = -f(x)$.

$-f(x) = -\left(\frac{1}{4}x^3 + x^2 - x - 3\right)$ $-f(x) = -\frac{1}{4}x^3 - x^2 + x + 3$

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

Conclusion

Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

Would you like further clarification on any part of this process? Here are some related questions for you:

1. What is the definition of an even function?
2. How can you check if a function is odd?
3. How do you find $f(x)$ and $f(-x)$ for higher-order polynomials?
4. Can a function be both even and odd?
5. What is the symmetry of a function and how does it relate to even and odd functions?

Tip: Always check the signs of the terms when substituting $-x$ into the function. The symmetry will often become clear through these changes.