$f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$
What is the domain of $(f\circ g)(x)$
I thought that since the root function must be greater than or equal to 0 in order to be a real number, I would calculate domain by determining where the radicand is greater than or equal zero and then excluding it:
$x-1\geqslant 0 \Longrightarrow x=1$
So, I thought the domain would therefore be $[1,\infty)$
However, my textbook solutions section says the domain is actually $(-\infty, \infty)$.
Why is that?
Best Answer
You don't need $x-1\geqslant 0$. It is $\sqrt[3]{x-1}$(which range and domain are $\mathbb R$), not $\sqrt{x-1}$(whose domain is $[1,\infty)$ and range is $[0,\infty)$).