Composite Function Domains

**superduper** · September 15th 2009, 12:51 PM

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

7. Answer with brief Explanation or Counterexample. Is the domain of $f \circ g$ contained in the domain of $f$ ?

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

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

What do you guys think?

**HallsofIvy** · September 16th 2009, 04:39 AM

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 $f$ and the domain of $g$ , but is this correct? And is this an acceptable way of showing that answer?

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

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 $f\circ g$ 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 $g\circ f$ is $f^{-1}(dom(g))$ . The domain of $f\circ g$ is $g^{-1}(dom(f))$ .

The answer to the question given, is "No, the $f\circ g$ 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)= $\sqrt{x}$ . $g\circ f(x)= \sqrt{-x}$ 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.

**superduper** · September 16th 2009, 06:26 AM

Perfect, that makes a lot more sense. Thanks man

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