# If $f(x)=x^3+1$ and $g(x)=\sqrt[3]{x-1}$ then what is the domain of $(f\circ g)(x)$

$$f(x)=x^3+1$$ 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?

You don't need $$x-1\geqslant 0$$. It is $$\sqrt[3]{x-1}$$(which range and domain are $$\mathbb R$$), not $$\sqrt{x-1}$$(which domain is $$[1,\infty)$$ and range is $$[0,\infty)$$).