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**Composite functions [SL/HL]**

If we have a function *f *(*x*) and a function *g*(*x*)* * then the function

*y*=*f *(*g *(*x*))

is the function we get by first applying the function *g* then the function *f*.

The expression *f *(*g *(*x*)) is read “*f* of *g* of *x*“.

One can often see this written as

y=(*f* ∘ *g*)(*x*)

Say we have *f *(*x*)=2*x*+3 and *g* (*x*)=*x*^{2}+*x*, then we have that

*f *(*g *(*x*))=*f *(*x ^{2}+x*)=2(

*x*

^{2}+

*x*)+3=2

*x*

^{2}+2

*x*+3

We thus replace *g *(*x*) by *x ^{2}+x*, then use that instead of

*x*in

*f*(

*x*)=2

*x*+3.

If we instead look at

*g** (f (x))=g *(*2x+3*)=(*2x+3*)^{2}+(*2x+*3)

=4*x*^{2}+12*x*+9+*2x+*3*
*=4

*x*

^{2}+14

*x*+12

then we get a different result. We thus have that *f *(*g *(*x*)) is usually not equal to

*g **(f (x)).*

If you are given a value for *x*, say if you are to find *f *(*g*(2)) then don’t spend time creating the composite function first, unless you need it later on. Instead calculate *g *(2) = 2^{2}+2 =6, then you calculate *f* (6)=2·6+3=15. This is way faster.

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