Given the following functions
[tex]\begin{gathered} f(x)=3x-12 \\ g(x)=\frac{x}{3}+4 \end{gathered}[/tex]
We want to find their compositions. Evaluating a function means finding the value of the function that corresponds to a given value of x. Compose two functions is the same as "evaluating" the inside function on the outside function. To do this, simply replace all the x variables with the expression of the inside function.
For the first composition, we have
[tex]f(g(x))=f(\frac{x}{3}+4)=3(\frac{x}{3}+4)-12=x+12-12=x[/tex]
There are no restrictions on this function, therefore, the domain is all real numbers.
For the second composition, we have
[tex]g(f(x))=g(3x-12)=\frac{(3x-12)}{3}+4=x-4+4=x[/tex]
There are no restrictions on this function, therefore, the domain is all real numbers.
Since
[tex]f(g(x))=g(f(x))=x[/tex]
Those functions are the inverse of each other.
