1. FUNCTIONS – Composite Functions RULES :

2. FUNCTIONS – Composite Functions RULES : Read as f at g of x

3. FUNCTIONS – Composite Functions RULES : Read as f at g of x Read as g at f of x

4. FUNCTIONS – Composite Functions RULES : Read as f at g of x Read as g at f of x Symbol DOES NOT mean multiply !!!

5. FUNCTIONS – Composite Functions RULES : Read as f at g of x Read as g at f of x It is the symbol used to show composite…

6. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( ƒ g )(3) if ƒ(x) = 2x – 6 and g(x) = x 2

7. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( ƒ g )(3) if ƒ(x) = 2x – 6 and g(x) = x 2 Using the Rule

8. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( ƒ g )(3) if ƒ(x) = 2x – 6 and g(x) = x 2 Using the Rule

9. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( ƒ g )(3) if ƒ(x) = 2x – 6 and g(x) = x 2 Using the Rule We need to find g(3) first

10. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( ƒ g )(3) if ƒ(x) = 2x – 6 and g(x) = x 2 Using the Rule Now we place 9 into f(x)

11. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( g ƒ )(-2) if ƒ(x) = x 2 – x – 12 and g(x) = 0.5x + 4 Using the Rule :

12. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( g ƒ )(-2) if ƒ(x) = x 2 – x – 12 and g(x) = 0.5x + 4 Using the Rule : First find ƒ( -2 )

13. FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work inside out EXAMPLE : Find ( g ƒ )(-2) if ƒ(x) = x 2 – x – 12 and g(x) = 0.5x + 4 Using the Rule : Now substitute -6 in g(x)

14. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4

15. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4 Substitute into x

16. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4 Substitute into x

17. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4 Combined like terms

18. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x 2 – 2x + 8

19. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x 2 – 2x + 8 Substituted ( x + 3 ) for all xs

20. RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x 2 – 2x + 8

21. Composite Functions : going backwards Suppose we were given a function h(x) that was the result of a composite operation. How could we determine the two functions that were combined to get that function ? We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses or under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something.

22. Composite Functions : going backwards We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something. Example : If h(x) = ( x + 2 ) 2 was created by ( f g )(x) find the functions f(x) and g(x) that created h(x)

23. Composite Functions : going backwards We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something. Example : If h(x) = ( x + 2 ) 2 was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x + 2 ) 2 ** you can see, that x + 2 is raised to the 2 nd power ** so x + 2 is our something raised to a power

24. Composite Functions : going backwards We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something. Example : If h(x) = ( x + 2 ) 2 was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x + 2 ) 2 ** you can see, that x + 2 is raised to the 2 nd power ** so x + 2 is our something raised to a power Therefore :g ( x ) = x + 2

25. Composite Functions : going backwards We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something. Example : If h(x) = ( x + 2 ) 2 was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x + 2 ) 2 ** you can see, that x + 2 is raised to the 2 nd power ** so x + 2 is our something raised to a power Therefore :g ( x ) = x + 2 Removing the x + 2 from the parentheses and replacing it with just x creates our f(x)

26. Composite Functions : going backwards We will always use h(x) = ( f g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What youll see is something raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the something. Example : If h(x) = ( x + 2 ) 2 was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x + 2 ) 2 ** you can see, that x + 2 is raised to the 2 nd power ** so x + 2 is our something raised to a power Therefore :g ( x ) = x + 2 AND f(x) = ( x ) 2 Removing the x + 2 from the parentheses and replacing it with just x creates our f(x)

27. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x)

28. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) Can you see the something ??

29. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) Can you see the something ?? Its x – 5

30. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) Can you see the something ?? Its x – 5 Therefore : g ( x ) = x – 5

31. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 Now replace the x – 5 inside each parentheses with just x

32. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 Now replace the x – 5 inside each parentheses with just x f ( x ) = 4( x ) 3 + 2( x )

33. Composite Functions : going backwards Example : If h(x) = 4( x – 5 ) 3 + 2( x – 5 ) was created by ( f g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 f ( x ) = 4( x ) 3 + 2( x )

34. Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the something ??

35. Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the something ?? 3x – 10 is under a root

36. Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the something ?? 3x – 10 is under a root So : g ( a ) = 3x - 10

37. Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) g ( x ) = 3x – 10 Now replace the 3x – 10 under the root with just x

38. Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) g ( x ) = 3x – 10 f ( x ) = Now replace the 3x – 10 under the root with just x