Question

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

Answer 1

Step 1: Define the composite function

1. When two or more functions are combined together in a single function, the process is known as the composition of function.
2. The considered single function is known as the composite function.
3. Let, $f\left(x\right)$ and $g\left(x\right)$ be two individual functions, then $f\left(g\left(x\right)\right)$ is a composite function, where $x$ is the input of $g\left(x\right)$ and $g\left(x\right)$ is the input of the function $f\left(x\right)$, it can be also denoted by, $\left(f\circ g\right)\left(x\right)$.

Step 2: Example of a composite function

Let us consider two individual functions, $f\left(x\right)=\frac{2}{2x+3}$ and $g\left(x\right)=x^2+1$.

Then the composite function $f\left(g\left(x\right)\right)$ can be given by,

$$\begin{gathered} f\left(g\left(x\right)\right)=f\left(x^2+1\right) \\ \Rightarrow f\left(g\left(x\right)\right)=\frac{2}{2\left(x^2+1\right)+3} \\ \Rightarrow f\left(g\left(x\right)\right)=\frac{2}{2x^2+2+3} \\ \Rightarrow f\left(g\left(x\right)\right)=\frac{2}{2x^2+5} \end{gathered}$$

Therefore, the composite function can be given by, $\mathit{f}\left(g\left(x\right)\right)\mathbf{=}\frac{\mathbf{2}}{\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{5}}$.