1. ## Composite Function Domains

Hey, I'm having trouble trying to figure out this general word problem.

7. Answer with brief Explanation or Counterexample. Is the domain of $\displaystyle f \circ g$ contained in the domain of $\displaystyle f$?
I want to say that the domain of the composite should be contained within the intersection of the range of $\displaystyle f$ and the domain of $\displaystyle g$, but is this correct? And is this an acceptable way of showing that answer?

$\displaystyle D_{f \circ g}=E_{f} \cap D_{g}$ where $\displaystyle E_{f}=range\ of\ f$ and $\displaystyle D_{g}=domain\ of\ g$

What do you guys think?

2. Originally Posted by superduper
Hey, I'm having trouble trying to figure out this general word problem.

I want to say that the domain of the composite should be contained within the intersection of the range of $\displaystyle f$ and the domain of $\displaystyle g$, but is this correct? And is this an acceptable way of showing that answer?

$\displaystyle D_{f \circ g}=E_{f} \cap D_{g}$ where $\displaystyle E_{f}=range\ of\ f$ and $\displaystyle D_{g}=domain\ of\ g$

What do you guys think?
Almost. What you give is a subset of the range of f and I believe you want the domain. The domain of $\displaystyle f\circ g$ is the subset of the domain of f that is mapped into that intersection. And since that is necessarily a subset of the range of f, you don't need to say that. Oh, and you have the order of the functions reversed. The domain of $\displaystyle g\circ f$ is $\displaystyle f^{-1}(dom(g))$. The domain of $\displaystyle f\circ g$ is $\displaystyle g^{-1}(dom(f))$.

The answer to the question given, is "No, the $\displaystyle f\circ g$ is not necessarily contained in the domain of f, it is contained in the domain of g". A counterexample to the original statement is g(x)= -x, f(x)= $\displaystyle \sqrt{x}$. $\displaystyle g\circ f(x)= \sqrt{-x}$ so its domain is the set of all non-[b]positive[/tex] numbers while the domain of f is the set of all non-negative numbers.

3. Perfect, that makes a lot more sense. Thanks man