Question

State the domain of f•g.

Answer 1

Answer:  choice A

$D: x > -3; \ (f \circ g)(x) = \frac{-7}{\sqrt{x+3}}$

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Explanation:

The idea is to start with f(x), then replace every copy of x with g(x) so that you end up with the composition function as shown below.

$f(x) = \frac{-7}{x}\\\\f(g(x)) = \frac{-7}{g(x)}\\\\f(g(x)) = \frac{-7}{\sqrt{x+3}}\\\\(f \circ g)(x) = \frac{-7}{\sqrt{x+3}}$

The notation $f(g(x))$ is the same as $(f \circ g)(x)$

I prefer $f(g(x))$ as it is more descriptive.

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Now onto computing the domain. We want to avoid dividing by zero, and we want to avoid taking the square root of a negative number.

This means the expression x+3 must be larger than 0. That way $\sqrt{x+3}$ will also be larger than 0.

If x+3 > 0, then x > -3 after subtracting 3 from both sides.

As long as x is larger than -3, then it is a valid input to the composite function $(f \circ g)(x)$