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Block 3
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
Composite functions f(g(x)) or
f(g(x))
Change this
g(f(x))
f(x)= 2x + 1 g(x) = 4x
Composite functions f(g(x)) or
f(g(x))
= f(4x)
Change this
g(f(x))
f(x)= 2x + 1 g(x) = 4x
Composite functions f(g(x)) or
f(g(x))
= f(4x)
Change this
= 2(4x) + 1
= 8x + 1
g(f(x))
f(x)= 2x + 1 g(x) = 4x
Composite functions f(g(x)) or
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
Composite functions f(g(x)) or
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
Composite functions f(g(x)) or
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
Composite functions f(g(x)) or
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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Composite functions

  • 2. What is to be Learned • What a composite function is • How to express a composite function
  • 3. Definition • A number (or letter) is affected by more than one function
  • 4. 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
  • 5. f(x) = x2 f(4) = 16 f(16) = 162 f( ) ?f(4) f( )f(4)16
  • 6. f(x) = 3x + 2 f(4) = 14 f(14) = 3(14) + 2 f( ) ?f(4) f( )f(4)14
  • 7. 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
  • 8. With Formulas f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) g(f(x)) Change this
  • 9. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) Change this g(f(x))
  • 10. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) = f(4x) Change this g(f(x))
  • 11. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) = f(4x) Change this = 2(4x) + 1 = 8x + 1 g(f(x))
  • 12. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) = f(4x) Change this = 2(4x) + 1 = 8x + 1 g(f(x)) g(f(x))
  • 13. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) = f(4x) = 2(4x) + 1 = 8x + 1 g(f(x)) = g(2x + 1) g(f(x))
  • 14. f(x)= 2x + 1 g(x) = 4x Composite functions f(g(x)) or f(g(x)) = f(4x) = 2(4x) + 1 = 8x + 1 g(f(x)) = g(2x + 1) = 4(2x +1) = 8x + 4 g(f(x))
  • 15. f(x)= 3x + 4 g(x) = 2x Composite functions f(g(x)) or f(g(x)) = f(2x) g(f(x)) = 3(2x) + 4 = 6x + 4 g(f(x)) = g(3x + 4) = 2(3x +4) = 6x + 8
  • 16. f(x)= x2 – 4 g(x) = 3x Find f(g(x)) f(g(x)) = f(3x) = (3x)2 – 4 = 9x2 – 4 Key Question