Question

Suppose the domain of g is (negative infinity, infinity). Is the domain of f o g( negative infinity, infinity)? Explain why of why not.

Please heelp!

Answer 1

Not in general, no. As a counter-example consider
\[f(x)=\frac{1}{x}, \text{ and }g(x)=x.\]
Then,
\[(f\circ g)(x)=f(g(x))=f(x)=\frac{1}{x}.\]
As you can see, the domain of g is (-infinity, infinity), whereas the domain of f is (-infinity, 0) U (0, infinity). Thus, the domain of (f o g) is the same as the domain of f.