Question

...Determine whether f(x) and g(x) are inverses of each other.
f(x)=(2)/(3)x+6
g(x)=(3)/(2)x-9

Answer 1

We have
\[f(x)=\frac{2}{3} x+6\]
\[g(x)=\frac{3}{2} x-9\]
If f(x) and g(x) are inverse of each other then
\[f(g(x))=x\]
Let's check for this
\[f(g(x))=\frac{2}{3} g(x)+6\]
\[g(x)=\frac{3}{2} x-9\]
Now
\[f(g(x))=\frac{2}{3} (\frac{3}{2} x-9)+6=x-\frac{2}{3} \times 9+6=x-6+6=x\]
Hence f(x) and g(x) are inverse of each other