Find the domain of composition of two functions

Question

Let $$f(x)= \frac{x}{x-9} \text{ and } g(x) = \frac{-6}{x+5} $$ Find the domain of $f \circ g.$

Please show the steps and substitution/algebra to the problem. Thank you.

Answer 1

Guide:

• What is the domain of $g$? call it $D$.

• What is the domain of $f$? there is a value of that is being excluded from the real value, call that number $c$.

• Solve for $y$ in $g(y)=c$, you have to make sure $y$ is not in the domain of $f \circ g$.

• The domain is $D \setminus \{ y\}$.

Answer 2

$f\circ g(x)=\frac{g(x)}{g(x)-9}=\frac{\frac {-6}{x+5}}{\frac{-6}{x+5}-9}=\frac{\frac {-6}{x+5}}{\frac{-6-9x-45}{x+5}}=\frac 6{9x+51}=\frac2{3x+17}$

$\therefore x\not=\frac{-17}3, -5 $...

Actually, I guess I did this "the long way". We can't have $x=-5$ because it's not in the domain of $g$...