1. **Problem:** Determine if the function $f(x) = x^{-1} + x^7 + x^{-3}$ is even, odd, or neither. 2. **Recall definitions:** - A function $f$ is **even** if $f(-x) = f(x)$ for all $x$ in the domain. - A function $f$ is **odd** if $f(-x) = -f(x)$ for all $x$ in the domain. - Otherwise, the function is **neither**. 3. **Evaluate $f(-x)$:** $$f(-x) = (-x)^{-1} + (-x)^7 + (-x)^{-3} = -x^{-1} - x^7 - x^{-3}$$ 4. **Compare $f(-x)$ with $f(x)$:** - $f(-x) = -x^{-1} - x^7 - x^{-3}$ - $-f(x) = -(x^{-1} + x^7 + x^{-3}) = -x^{-1} - x^7 - x^{-3}$ Since $f(-x) = -f(x)$, the function is **odd**. --- **Final answer:** $f(x) = x^{-1} + x^7 + x^{-3}$ is **odd**.