Composite functions

What is to be Learned
• What a composite function is
• How to express a composite function

Definition
• A number (or letter) is affected by more
than one function

f(x) = 3x and g(x) = x2
Combining these:
Multiply by 3, then square or
Square, then multiply by 3
Not the same!
Multiply by 3 Square

f(x) = x2
f(4) = 16
f(16) = 162
f( ) ?f(4)
f( )f(4)16

f(x) = 3x + 2
f(4) = 14
f(14) = 3(14) + 2
f( ) ?f(4)
f( )f(4)14

Combination of more than one function.
If f(x) = 4x and g(x) = x2
g( f(3) )?
f(3) = 12
so g( f(3) )
= g (12 ) =
= 144
122
Composite Functions

With Formulas
f(x)= 2x + 1 g(x) = 4x
Composite functions f(g(x)) or
f(g(x))
g(f(x))
Change this

f(x)= 2x + 1 g(x) = 4x
f(g(x))
Change this
g(f(x))

f(x)= 2x + 1 g(x) = 4x
f(g(x))
= f(4x)
Change this
g(f(x))

f(x)= 2x + 1 g(x) = 4x
f(g(x))
= f(4x)
Change this
= 2(4x) + 1
= 8x + 1
g(f(x))

f(x)= 2x + 1 g(x) = 4x
f(g(x))
= f(4x)
Change this
= 2(4x) + 1
= 8x + 1
g(f(x))
g(f(x))

f(x)= 2x + 1 g(x) = 4x
f(g(x))
= f(4x) = 2(4x) + 1
= 8x + 1
g(f(x))
= g(2x + 1)
g(f(x))

f(x)= 2x + 1 g(x) = 4x
f(g(x))
= f(4x) = 2(4x) + 1
= 8x + 1
g(f(x))
= g(2x + 1) = 4(2x +1)
= 8x + 4
g(f(x))

f(x)= 3x + 4 g(x) = 2x
f(g(x))
= f(2x)
g(f(x))
= 3(2x) + 4
= 6x + 4
g(f(x))
= g(3x + 4) = 2(3x +4)
= 6x + 8

f(x)= x2
– 4 g(x) = 3x
Find f(g(x))
f(g(x))
= f(3x) = (3x)2
– 4
= 9x2
– 4
Key Question

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