Determine algebraically if f(x) =5x^2_8x_x+1 is even, odd or neither

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.

Determine algebraically if f(x) = 5x^2 - 8x - x + 1 is even, odd or neither.

Learn how to algebraically determine if the function f(x) = 5x^2 - 9x + 1 is even, odd, or neither. This problem involves symmetry tests using key algebraic concepts such as even and odd functions, making it suitable for students in Grades 10-12.

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