Even Odd Functions 3B6D80

1. **State the problem:** Determine whether the functions $$h(x)=3x^3+5$$, $$f(x)=x+|x|$$, and $$h(x)=\frac{x}{x^2}-1$$ are even, odd, or neither. 2. **Recall definitions:** - A function $$g(x)$$ is **even** if $$g(-x) = g(x)$$ for all $$x$$. - A function $$g(x)$$ is **odd** if $$g(-x) = -g(x)$$ for all $$x$$. - If neither condition holds, the function is **neither**. 3. **Analyze $$h(x) = 3x^3 + 5$$:** - Compute $$h(-x) = 3(-x)^3 + 5 = 3(-x^3) + 5 = -3x^3 + 5$$. - Compare with $$h(x)$$: - $$h(-x) \neq h(x)$$ (since $$-3x^3 + 5 \neq 3x^3 + 5$$). - $$h(-x) \neq -h(x)$$ (since $$-3x^3 + 5 \neq -(3x^3 + 5) = -3x^3 - 5$$). - Therefore, $$h(x)$$ is **neither** even nor odd. 4. **Analyze $$f(x) = x + |x|$$:** - Compute $$f(-x) = (-x) + |-x| = -x + |x|$$. - Compare with $$f(x)$$: - $$f(x) = x + |x|$$. - Check if even: - Is $$f(-x) = f(x)$$? - $$-x + |x| \stackrel{?}{=} x + |x|$$ - This is false for general $$x$$. - Check if odd: - Is $$f(-x) = -f(x)$$? - $$-x + |x| \stackrel{?}{=} -(x + |x|) = -x - |x|$$ - This is false for general $$x$$. - Therefore, $$f(x)$$ is **neither** even nor odd. 5. **Analyze $$h(x) = \frac{x}{x^2} - 1 = \frac{x}{x^2} - 1 = \frac{x}{x^2} - 1$$:** - Simplify $$h(x)$$: - $$h(x) = \frac{x}{x^2} - 1 = \frac{x}{x^2} - 1 = \frac{1}{x} - 1$$ (since $$\frac{x}{x^2} = \frac{1}{x}$$ for $$x \neq 0$$). - Compute $$h(-x) = \frac{1}{-x} - 1 = -\frac{1}{x} - 1$$. - Compare with $$h(x)$$: - Check if even: - $$h(-x) = h(x)$$? - $$-\frac{1}{x} - 1 \stackrel{?}{=} \frac{1}{x} - 1$$ - False. - Check if odd: - $$h(-x) = -h(x)$$? - $$-\frac{1}{x} - 1 \stackrel{?}{=} -\left(\frac{1}{x} - 1\right) = -\frac{1}{x} + 1$$ - False. - Therefore, $$h(x)$$ is **neither** even nor odd. **Final answers:** - $$h(x) = 3x^3 + 5$$ is **neither**. - $$f(x) = x + |x|$$ is **neither**. - $$h(x) = \frac{x}{x^2} - 1$$ (simplified to $$\frac{1}{x} - 1$$) is **neither**.