Determine whether each function is even, odd, or neither. ƒ(x)=x⁴-5x+8

Verified step by step guidance

1. Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function is neither even nor odd.
2. Start by finding \( f(-x) \) for the given function \( f(x) = x^{4} - 5x + 8 \). Substitute \( -x \) into the function:
3. \[ f(-x) = (-x)^{4} - 5(-x) + 8 \]
4. Simplify each term: \( (-x)^{4} = x^{4} \) because an even power eliminates the negative sign, and \( -5(-x) = 5x \). So, \( f(-x) = x^{4} + 5x + 8 \).
5. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \( f(x) = x^{4} - 5x + 8 \) and \( -f(x) = -x^{4} + 5x - 8 \). Since \( f(-x) \) is neither equal to \( f(x) \) nor \( -f(x) \), the function is neither even nor odd.

Key Concepts

Even Functions

An even function satisfies the condition f(-x) = f(x) for all x in its domain. Graphically, even functions are symmetric about the y-axis. Common examples include functions with even powers of x, like x² or x⁴.

Odd Functions

An odd function satisfies the condition f(-x) = -f(x) for all x in its domain. These functions have rotational symmetry about the origin. Examples include functions with odd powers of x, such as x³ or x.

Testing Function Symmetry Algebraically

To determine if a function is even, odd, or neither, substitute -x into the function and simplify. Compare the result to f(x) and -f(x). If equal to f(x), the function is even; if equal to -f(x), it is odd; otherwise, it is neither.