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Terry was asked to determine whether $f(x)=x^{3}+\frac{1}{x}$ is even, odd, or neither. Here is his work:$\newline$Step $1$: Find expression for $f(-x)$$\newline$$$\begin{aligned} f(-x) & =(-x)^{3}+\frac{1}{(-x)} \\ & =-x^{3}-\frac{1}{x} \end{aligned}$$$\newline$Step $2$: Check if $f(-x)$ is equal to $f(x)$ or $-f(x)$$\newline$$-x^{3}-\frac{1}{x}$ isn't the same as $f(x)=x^{3}+\frac{1}{x}$ or$\newline$$$-f(x)=-x^{3}+\frac{1}{x} \text {. }$$$\newline$Step $3$: Conclusion$\newline$$f(-x)$ isn't equivalent to either $f(x)$ or $-f(x)$, so $f$ is neither even nor odd.$\newline$Is Terry's work correct? If not, what is the first step where Terry made a mistake?$\newline$Choose $1$ answer:$\newline$(A) Terry's work is correct.$\newline$(B) Terry's work is incorrect. He first made a mistake in Step $1$.$\newline$(C) Terry's work is incorrect. He first made a mistake in Step $2$.$\newline$(D) Terry's work is incorrect. He first made a mistake in Step $3$.