Question

Please help me figure out this problem

Answer 1

Answer:

$f(g(x))=f\left(-\dfrac{3x+21}{5}\right)=x\\\\g(f(x))=g\left(-\dfrac{5}{3}x-7\right)=x$

Step-by-step explanation:

Fill in the given expressions and simplify.

$f(g(x))=f\left(-\dfrac{3x+21}{5}\right)=-\dfrac{5}{3}\left(-\dfrac{3x+21}{5}\right)-7\\\\=\left(-\dfrac{5}{3}\right)\left(-\dfrac{3x}{5}\right)-\left(\dfrac{5}{3}\right)\left(-\dfrac{21}{5}\right)-7\\\\=\dfrac{15x}{15}+\dfrac{5\cdot 21}{3\cdot 5}-7=x+7-7=x$

Going the other way, we have ...

$g(f(x))=\displaystyle -\frac{3\left(-\frac{5}{3}x-7\right)+21}{5}=-\frac{-5x-21+21}{5}=x$