Given that $f(x)$ is odd, I need to find if $f(\sec x)$ is odd.
We (or I, at least) have always defined an odd function in the following way: $f(x)$ is odd if $f(x)=-f(-x)$, for all $x$ in the domain of $f(x)$.
Since $f(\sec x)$ is defined in terms of $\sec x$ should I be checking if $f(\sec x)=f(-\sec x)$ or $f(x)=f(-x)$?
I think the former, but not according to my textbook — and there's no solution given to this problem so I'm not convinced why.
Any help would be appreciated.
Actually, the definition of odd function is: for each $x\in D_f$, $f(-x)=-f(x)$. In particular, since your function is $f\circ\sec$, asserting that it is odd means that$$(\forall x\in D_{f\circ\sec}):f\bigl(\sec(-x)\bigr)=-f\bigl(\sec(x)\bigr).$$But, in fact, your function is even, since $\sec$ is even, and therefore, if $x\in D_{f\circ\sec}$, then $f\bigl(\sec(-x)\bigr)=f\bigl(\sec(x)\bigr)$.