## Calculus (3rd Edition)

$$\cos \left(x^{-1}\right)$$ Domain: $\{x: x \neq 0\}$ $$\ (\cos x)^{-1}$$ Domain: $\{x: x \neq (2 k+1) \frac{\pi}{2}\}$
We are given the functions: $$h(x)=\cos x \text { and } g(x)=x^{-1}$$ We find the composite function as: \begin{align*} (h \circ g)(x)&=h(g(x))\\ &=h\left(x^{-1}\right)\\ &=\cos \left(x^{-1}\right) \end{align*} The domain is $\{x: x \neq 0\}$ Next, we find: \begin{align*} (g \circ h)(x)&=g(h(x))\\ &=g(\cos x)\\ &=(\cos x)^{-1} \end{align*} Domain is $\{x: \cos x \neq 0\}$, so $\{x: x \neq (2 k+1) \frac{\pi}{2}\}$