Question

HELP! Medal to whoever gets it.
https://aoa.sooschools.com/media/g_alg02_ccss_2016/1/g_alg02u01t19_q44_ps.gif
A. Use composition to prove whether or not the functions are inverses of each other.
B. Express the domain of the compositions using interval notation.

Answer 1

\[f(g(x)) = \frac{ 1 }{ g(x) - 5} = \frac{ 1 }{\frac{ 5x + 1 }{ x} - 5 } = \frac{ 1 }{ \frac{ 1 }{ x } } = x\]

\[g(f(x)) = \frac{ 5f(x) + 1 }{ f(x) } = \frac{ 5(\frac{ 1 }{ x-5 }) + 1 }{ \frac{ 1 }{ x-5 } }\]

Try simplifying the second one on your own.

Answer 2

Well, now that you now that they are the inverse of each other it should not be a problem to give the domain of the compositions.