YOU ARE LEARNING:
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)=x2. 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)=x2
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 x2 from g(x)=x2.
Change x to x2 on the right of f(x)=2x
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)=2×32.
For fg(x)=2x2, 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)=x2 with f(x)=2x.
Replace x with 2x in g(x)=x2
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)=4x2.
For gf(x)=4x2, what is gf(2)?
Let's try another example. We'll start with the functions below:
f(x)=2x−4 and g(x)=6x
Start with fg(x)
Remember, since fis first, we replace the x in f(x)=2x−4 with g(x).
What is fg(x)?
Now we can find outputs for fg(x)=26x−4
To find an output, we substitute the input for x.
What is fg(3)=26x−4?
Now let's try gf(x)
This time, we need to substitute the x in g(x)=6x with f(x)=2x−4.
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.
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)?
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)=x2. What is x if ff(x)=256?