### Composite Function

Functions can have almost any input. This includes other functions.

In the function notation notes we saw that a function such as $g(x)=3x^2+1$ could have an input of 2x which can be written in function notation as $g(2x)$. The input of 2x can be viewed as another function. Lets define $h(x)=2x$. We can then write the function *g* to have the an input of *h(x)*:

While the notation might be a little intimidating the statement $g(h(x))$ is equivalent to $g(2x)$. When the argument of one function is another function (i.e. the input of one function is a function) it is referred to as a composite function. The notation for a composite function can be written in two ways, both of which you should be familiar with:

(2)I personally don't like the second form as I find it less clear, but it will show up on homework, IB exams and beyond…

### Examples

Looking at the example above, we can write the composite of *g* and *h* in terms of *x*:

We can also make a composition of *h* as a function of *g*:

Note that in general $g(h(x)) \neq h(g(x))$. This is due to order of operations and using non-linear equations.

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