Composite Functions

Prerequisite Material:

Functions

Composite Functions:

Definition: a function made of up of two or more other functions, where the output of one is the input to the other and so on.

Video
from corbettmaths.com

Example:
If $f\left(x\right)=x+2$, then $f\left(3\right)=3+2=5$.
We could take this output and enter it into another function, say, $g\left(x\right)=3x-4$.
Entering the value from $f\left(x\right)$, then $g\left(5\right)=3\times 5-4=11$.

The same result can be found by substituting $f\left(x\right)$ into $g\left(x\right)$

$g\left(f\left(x\right)\right)=g(x+2)=3(x+2)-4=3x+2$

So, if $x=3$ then $g\left(f\left(3\right)\right)=3\times 3+2=11$.
That is the same as before.

An alternate way of writing $g\left(f\left(x\right)\right)$ can be written as $gf\left(x\right)$.
Note: the function $f$ is worked out before $g$.

In general $gf\left(x\right)$ is not the same as $fg\left(x\right)$.

Questions:

Questions
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Textbook Questions from corbettmaths.com
GCSE Questions & Answers from piacademy.co.uk
Further Maths Video from corbettmaths.com
Further Maths Practice Questions from corbettmaths.com

Further Reading:

Inverse Functions