Algebra

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Composite Functions

Composite functions enable us to perform two functions on a number at the same time.

We can perform multiple functions on an input. When we do this, it's called a composite function.

Let's start with two functions: $f(x)=2x$ and $g(x)=x^2$ . To form a composite functions, we can put these functions together.

The composite function will be named in the form $fg(x)$ , or $gf(x)$ . The order of the letters depends on the function that is performed first. The function that comes second, such as $g(x)$ in $fg(x)$ , is placed into the first function, in this case $f(x)$ . Let's have a look at this in more detail.

We'll start with our two functions:
$f(x) = 2x$ and $g(x) = x^2$

Let's find $fg(x)$

This means that we need to combine the two functions together. In this case, $f(x)$ comes first, which means we can replace the $x$ in $f(x)=2x$ with $x^2$ from $g(x)=x^2$ .

Change $x$ to $x^2$ on the right of $f(x)=2x$

$fg(x)=2x^2$

Now we have our composite function

We can use this to produce outputs in the same way as a normal function. All we need to do is replace $x$ with our input. For example, using $3$ as the input, the composite function would be $fg(3)=2\times3^2$ .

For $fg(x)=2x^2$ , what is $fg(3)$ ?

Now let's find $gf(x)$

This time, $g(x)$ is the first function. Therefore, we need to replace the $x$ in $g(x)=x^2$ with $f(x)=2x$ .

Replace $x$ with $2x$ in $g(x)=x^2$

Since the whole of $x$ is squared, the composite function is $gf(x)=(2x)^2$ . Therefore, we need to square everything in the bracket, leaving us with $gf(x)=4x^2$ .

For $gf(x)=4x^2$ , what is $gf(2)$ ?

Let's try another example. We'll start with the functions below:

$f(x) = \dfrac{x - 4}{2}$ and $g(x)=6x$

Start with $fg(x)$

Remember, since $f$ is first, we replace the $x$ in $f(x)=\dfrac{x-4}{2}$ with $g(x)$ .

What is $fg(x)$ ?

Now we can find outputs for $fg(x)=\dfrac{6x-4}{2}$

To find an output, we substitute the input for $x$ .

What is $fg(3)=\dfrac{6x-4}{2}$ ?

Now let's try $gf(x)$

This time, we need to substitute the $x$ in $g(x)=6x$ with $f(x)=\dfrac{x-4}{2}$ .

What is $gf(x)$ ?

If $g(x) =2 - x$ and $f(x) = 2x + 4$ what is $fg(x)$ ? Give your answer in its simplest form.

We can also use functions and composite functions in algebraic problems.

$g(x) = 2x + 3$

What is $x$ if $gg(x) = 49$ ?

The first step is to construct $gg(x)$

We need to replace $x$ in $gg(x)=2x+3$ with $g(x)=2x+3$ .

What is $gg(x)$ ?

$gg(x)=2(2x+3)+3$

We can simplify this to $gg(x)=4x+9$ . From the question, we know that $gg(x)=4x+9=49$ , so we can conclude that $4x+9=49$ .

If $4x+9=49$ , what is $x$ ?

If $f(x) = x^2$ . What is $x$ if $ff(x) = 256$ ?