To determine whether the function $f(x) = 5x^2 - 8x - x + 1$ is even, odd, or neither, we need to follow these steps:

Step 1: Simplify the function

First, simplify the given function: $f(x) = 5x^2 - 8x - x + 1$ Combine like terms: $f(x) = 5x^2 - 9x + 1$

Step 2: Test for Even Function

A function is even if $f(-x) = f(x)$.

Substitute $-x$ into the function: $f(-x) = 5(-x)^2 - 9(-x) + 1$ $f(-x) = 5x^2 + 9x + 1$

Now compare $f(-x) = 5x^2 + 9x + 1$ with $f(x) = 5x^2 - 9x + 1$. Since $f(-x) \neq f(x)$, the function is not even.

Step 3: Test for Odd Function

A function is odd if $f(-x) = -f(x)$.

We already have $f(-x) = 5x^2 + 9x + 1$.

Now calculate $-f(x)$: $-f(x) = -(5x^2 - 9x + 1) = -5x^2 + 9x - 1$

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

Conclusion:

Since the function is neither even nor odd, we conclude that: $f(x) = 5x^2 - 9x + 1 \text{ is neither even nor odd.}$

Would you like further clarification or details on this?

Relative questions:

1. How do you check if a function is even algebraically?
2. What are the properties of odd functions?
3. Can a function be both even and odd?
4. How does the degree of a polynomial relate to its symmetry properties?
5. What role does the constant term play in determining if a function is even or odd?

Tip:

To quickly identify whether a polynomial function might be even or odd, check the exponents of each term. Functions with all even powers of $x$ tend to be even, and functions with all odd powers of $x$ tend to be odd, but this is not a strict rule when constants are involved.