# Composite functions

**What are composite functions**

A composite function is a function created by exchanging one function into another function. If there are two functions $$f(x)$$ and $$g(x)$$, the representation of the composite function is $$f {\circ}{g}(x)$$ or $$g {\circ}{f}(x)$$, where $$\circ$$ is the composition symbol. In this, $$f {\circ}{g}(x)$$ and $$g {\circ}{f}(x)$$ are the composite functions of $$f(x)$$ and $$g(x)$$. Functions are assembled by replacing one function with another.

**Inverse Functions**

The inverse function with the aid of composition function $$f \circ{g}(x)$$ will be equal to $$x$$, and the value of the function $$g \circ{f}(x)$$ will be equal to $$x$$. When the value of both functions $$f \circ{g}(x)$$ and $$g\circ{f}(x)$$ is equal to $$x$$, the function will be an inverse function.

**E2.9: Form composite functions defined by $$g(f(x))=g(f(x))$$.**

The order in which functions are composed offers a distinction in the result. If the order changes from $$f(g(x))$$ to $$g(f(x))$$, there will be a difference in the results. In general, the value of $$g{f(x)}$$ is not equal to the value of $$f{g(x)}$$. This is sometimes known as a function of a function. There are two forms of composite function:

One is the small circle form.

Another form is the nested form.

It is represented as $$f \circ{g}(x)=f(g(x))$$, which is read $$f$$ of $$g$$ of $$x$$. Or represented as $$g \circ{f}(x)=g(f(x))$$, which is read $$g$$ of $$f$$ of $$x$$. In this, $$f \circ{g}(x)$$ or $$g \circ{f}(x)$$ is the small circle form, and $$f(g(x))$$ or $$g(f(x))$$ is the nested form.

**In example**

If Muskan wants to go to Holland for a trip from Delhi, she needs to take a flight from Delhi to London and then London to Holland. So, consider a flight from Delhi to London $$f(x)$$ and flight from London to Holland $$g(x)$$. The direct relation from Delhi to Holland will be $$g(f(x))$$, and the relation from Holland to Delhi will be $$f(g(x))$$.

The function $$g{f(x)}$$ can be also be written as $$g(f(x))$$, i.e., the value of $$g{f(x)}$$ is the same as the value of $$g(f(x))$$.

**Worked examples**

**Example 1:** Find the value of $$gf(x)$$ for the functions $$f(x)=4x$$ and $$g(x)=x^{2}-4$$.

Step 1: Given information

Given functions, $$f(x)=4x$$ and $$g(x)=x^{2}-4$$

Step 2: Substitute $$x$$ with $$4x$$ in the function $$g(x)=x^{2}-4$$.

$$g \circ f(x)= (4x)^{2}-4$$

Step 3: Solve the function $$g\circf(x)=(4x)^{2}-4$$.

$$g \circ f(x)=16x^{2}-4$$

So, the value of $$g \circ f(x)$$ is $$16x^{2}-4$$.

**Example 2:** Find the value of $$g\circ f(x)$$ for functions $$f(x)=3x^{2}+5$$ and $$g(x)=5x-2$$.

Step 1: Given information

Given functions, $$f(x)= 3x^{2}+5$$ and $$g(x)=5x-2$$

Step 2: Substitute $$x$$ with $$3x^{2}+5$$ in the function $$g(x)=5x-2$$.

$$g \circ f(x)=5(3x^{2}+5)-2$$

Step 3: Solve the function $$g\circf(x)=5(3x^{2}+5)-2$$.

$$g \circ f(x)=15x^{2}+25-2$$

Step 4: Write the final value of $$g\circf(x)$$.

$$g \circ f(x)=15x^{2}+23$$

So, the value of $$g \circ f(x)$$ is $$15x^{2}+23$$.