QA,B: Each of the following functions f is bijective. Describe its inverse.
A: $$f:\mathbb{R} \rightarrow (0,\infty); \text{ defined by } f(x)=e^x $$
For this function, I said the inverse is: $$f^{-1}:(0,\infty) \rightarrow \mathbb{R}; \text{ defined by } f^{-1}(x)=\ln(x) $$
B:
$$f:\mathbb{R} \rightarrow \mathbb{R}; \text{ defined by } f(x)=x^3+1 $$
At first, for the inverse, I said:
$$f^{-1}:\mathbb{R} \rightarrow \mathbb{R}; \text{ defined by } f^{-1}(x)=\sqrt[3]{x-1}$$
but then I notice if x=-1 then the value inside the root will be negative which is illegal, therefore I adjusted by domain and codomain and obtained:
$$f^{-1}:\mathbb{R^{\geq 1}} \rightarrow \mathbb{R^{\geq 0}}; \text{ defined by } f^{-1}(x)=\sqrt[3]{x-1}$$
Can anyone verify if these answers are correct?
They are fine except you should not adjust the domain in the second example. It is perfectly fine to have a negative inside a cube root. For example, $\sqrt[3]{-64}=-4$.