Question

How do you find (f og)(x) and (g of)(x) given $f\left( x \right)=9x-5$ and $g\left( x \right)=9-3x$?

Answer 1

Hint: (f og)(x) is equal to f(g(x)). So, substitute g(x) in place of ‘x’ in the expression f(x) to find (f og)(x). Similarly, (g of)(x) is equal to g(f(x)). So, substitute f(x) in place of ‘x’ in the expression g(x) to find (g of)(x).

Complete step by step solution:
Given $f\left( x \right)=9x-5$ and $g\left( x \right)=9-3x$
(f og)(x) and (g of)(x) are the composite functions formed by the two functions f(x) and g(x).
As we know, (f og)(x) = f(g(x))
So, substituting g(x) into f(x), we get
$\begin{align}
  & \Rightarrow f(g(x))=9\cdot g(x)-5 \\
 & \Rightarrow f(g(x))=9(9-3x)-5 \\
 & \Rightarrow f(g(x))=81-27x-5 \\
 & \Rightarrow f(g(x))=76-27x \\
\end{align}$
Again, as we know (g of)(x) = g(f(x))
So, substituting f(x) into g(x), we get
$\begin{align}
  & \Rightarrow g(f(x))=9-3\cdot f(x) \\
 & \Rightarrow g(f(x))=9-3(9x-5) \\
 & \Rightarrow g(f(x))=9-27x+15 \\
 & \Rightarrow g(f(x))=24-27x \\
\end{align}$
This is the required solution of the given question.

Note:
Composite function (say h(x)) is a function whose values are found from two given functions (say f(x) and g(x)) by applying one function to an independent variable and then applying the second function to the result and whose domain consists of those values of the independent variable for which the result yielded by the first function lies in the domain of the second.
h(x) = f(g(x)) or g(f(x))
For the given question (f og)(x) and (g of)(x) are in the form of h(x).