1. Problem: Determine $f(-x)$ and $-f(-x)$ for each function and use these to decide if the function is even, odd, or neither. 2. Recall definitions: - Even function: $f(-x) = f(x)$ - Odd function: $f(-x) = -f(x)$ - Neither: if neither condition holds. 3. For each function: a) $f(x) = x^{2} - 4$ - Calculate $f(-x) = (-x)^{2} - 4 = x^{2} - 4$ - Calculate $-f(-x) = -(x^{2} - 4) = -x^{2} + 4$ - Compare: - $f(-x) = f(x)$, so $f$ is even. b) $f(x) = \sin x + x$ - Calculate $f(-x) = \sin(-x) + (-x) = -\sin x - x$ - Calculate $-f(-x) = -(-\sin x - x) = \sin x + x$ - Compare: - $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so $f$ is neither even nor odd. c) $f(x) = \frac{1}{x} - x$ - Calculate $f(-x) = \frac{1}{-x} - (-x) = -\frac{1}{x} + x$ - Calculate $-f(-x) = -(-\frac{1}{x} + x) = \frac{1}{x} - x$ - Compare: - $f(-x) \neq f(x)$ but $-f(-x) = f(x)$, so $f$ is odd. d) $f(x) = 2x^{3} + x$ - Calculate $f(-x) = 2(-x)^{3} + (-x) = -2x^{3} - x$ - Calculate $-f(-x) = -(-2x^{3} - x) = 2x^{3} + x$ - Compare: - $f(-x) \neq f(x)$ but $-f(-x) = f(x)$, so $f$ is odd. e) $f(x) = 2x^{2} - x$ - Calculate $f(-x) = 2(-x)^{2} - (-x) = 2x^{2} + x$ - Calculate $-f(-x) = -(2x^{2} + x) = -2x^{2} - x$ - Compare: - $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so $f$ is neither even nor odd. f) $f(x) = |2x + 3|$ - Calculate $f(-x) = |2(-x) + 3| = |-2x + 3|$ - Calculate $-f(-x) = -|-2x + 3|$ - Compare: - $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$ in general, so $f$ is neither even nor odd. Final classification: a) even b) neither c) odd d) odd e) neither f) neither