Question

Explain the concept of composite functions with the help of an example.

Answer 1

In this problem we need to explain the concept of the composite functions with the help of an example. For this we will first define the composite function after that we will consider any function and try to explore more on the composite function using those examples.

Complete step-by-step solution:
Composite function is a function which will depend on another function that means when a function is substituted in another function a composite function is created.
If $f\left( x \right)$, $g\left( x \right)$ are any two functions, then the composite function when we substituted the function $g\left( x \right)$ in the function $f\left( x \right)$ is represented by $f\left( g\left( x \right) \right)$. It can be termed as “$f$ of $g$ of $x$”. Also, when the function $f\left( x \right)$ is substituted in the function $g\left( x \right)$ the composite function formed is represented by $g\left( f\left( x \right) \right)$. It can be termed as “ $g$ of $f$ of $x$”.
Now let use consider that $f\left( x \right)=2x+1$, $g\left( x \right)=3x+5$.
Now the composite function $f\left( g\left( x \right) \right)$ is calculated by substituting $g\left( x \right)$ in $f\left( x \right)$, then we will get
$\begin{aligned} & f\left( g\left( x \right) \right)=f\left( 3x+5 \right) \\\ & \Rightarrow f\left( g\left( x \right) \right)=2\left( 3x+5 \right)+1 \\\ & \Rightarrow f\left( g\left( x \right) \right)=6x+10+1 \\\ & \Rightarrow f\left( g\left( x \right) \right)=6x+11 \\\ \end{aligned}$
Hence the composite function $f\left( g\left( x \right) \right)$ is given by $6x+11$ where $f\left( x \right)=2x+1$, $g\left( x \right)=3x+5$.
Now the composite function $g\left( f\left( x \right) \right)$ is calculated by substituting $f\left( x \right)$ in $g\left( x \right)$, then we will get
$\begin{aligned} & g\left( f\left( x \right) \right)=g\left( 2x+1 \right) \\\ & \Rightarrow g\left( f\left( x \right) \right)=3\left( 2x+1 \right)+5 \\\ & \Rightarrow g\left( f\left( x \right) \right)=6x+3+5 \\\ & \Rightarrow g\left( f\left( x \right) \right)=6x+8 \\\ \end{aligned}$
Hence the composite function $g\left( f\left( x \right) \right)$ is given by $6x+8$ where $f\left( x \right)=2x+1$, $g\left( x \right)=3x+5$.

Note: In this problem we have only considered a simple monomial as examples. We can also consider quadratic equations or cubic equations or and so on to form a composite function. In composite functions we need to remember one thing that the value of $f\left( g\left( x \right) \right)\ne g\left( f\left( x \right) \right)$.