1. The problem asks us to identify which function is odd from the given options. 2. Recall that a function $f(x)$ is odd if it satisfies the condition: $$f(-x) = -f(x)$$ for all $x$ in the domain. 3. Let's check each function: - a. $f(x) = |x|$ - $f(-x) = |-x| = |x|$ - Since $f(-x) = f(x)$, this function is even, not odd. - b. $f(x) = x^2$ - $f(-x) = (-x)^2 = x^2$ - Since $f(-x) = f(x)$, this function is even, not odd. - c. $f(x) = \cos x$ - $f(-x) = \cos(-x) = \cos x$ - Since $f(-x) = f(x)$, this function is even, not odd. - d. $f(x) = x^3$ - $f(-x) = (-x)^3 = -x^3 = -f(x)$ - This satisfies the odd function condition. 4. Therefore, the only odd function among the options is $f(x) = x^3$. **Final answer:** d. $f(x) = x^3$