1. Problem: For each function, find $f(-x)$ and $-f(-x)$, then compare with $f(x)$ to determine if the function is even, odd, or neither. 2. Function a) $f(x) = x^2 - 4$ - Compute $f(-x) = (-x)^2 - 4 = x^2 - 4$ - Compute $-f(-x) = -(x^2 - 4) = -x^2 + 4$ - Compare: - $f(-x) = f(x)$ so $f$ is even. 3. Function b) $f(x) = \\sin x + x$ - Compute $f(-x) = \\sin(-x) + (-x) = -\\sin x - x$ - Compute $-f(-x) = -(-\\sin x - x) = \\sin x + x$ - Compare: - $f(-x) eq f(x)$ and $-f(-x) = f(x)$ so $f$ is odd. 4. Function c) $f(x) = \frac{1}{x} - x$ - Compute $f(-x) = \frac{1}{-x} - (-x) = -\frac{1}{x} + x$ - Compute $-f(-x) = -(-\frac{1}{x} + x) = \frac{1}{x} - x$ - Compare: - $f(-x) eq f(x)$ and $-f(-x) = f(x)$ so $f$ is odd. 5. Function d) $f(x) = 2x^3 + x$ - Compute $f(-x) = 2(-x)^3 + (-x) = -2x^3 - x$ - Compute $-f(-x) = -(-2x^3 - x) = 2x^3 + x$ - Compare: - $f(-x) eq f(x)$ and $-f(-x) = f(x)$ so $f$ is odd. 6. Function e) $f(x) = 2x^2 - x$ - Compute $f(-x) = 2(-x)^2 - (-x) = 2x^2 + x$ - Compute $-f(-x) = -(2x^2 + x) = -2x^2 - x$ - Compare: - $f(-x) eq f(x)$ and $-f(-x) eq f(x)$ so $f$ is neither even nor odd. 7. Function f) $f(x) = |2x + 3|$ - Compute $f(-x) = |2(-x) + 3| = |-2x + 3|$ - Usually $|2x+3| \neq |-2x+3|$ and $-f(-x) = -| -2x + 3|$ which is negative whereas $f(x)$ is non-negative - So $f(-x) \neq f(x)$ and $-f(-x) \neq f(x)$, function is neither even nor odd. Final answers: - a) even - b) odd - c) odd - d) odd - e) neither - f) neither