The document discusses combining functions through addition, subtraction, multiplication, and division. Some key points: - Functions can be combined using the same rules as algebraic expressions, such as f(x) + g(x) = (f + g)(x). - An example demonstrates combining two functions f(x) and g(x) through addition, subtraction, multiplication, and division. - Composite functions are discussed where the output of one function acts as the input for another function, written as f(g(x)). - Several examples demonstrate evaluating composite functions by first evaluating the inner function and then the outer function. - It is shown that f(g(x)) is not

1. 2.7 Combining Functions
John 16:33 I have said these things to you, that in me you
may have peace. In the world you will have tribulation. But
take heart; I have overcome the world.”

2. We can add, subtract, multiply, divide functions

3. We can add, subtract, multiply, divide functions
f (x) + g(x) = ( f + g)(x) D : D f I Dg

4. We can add, subtract, multiply, divide functions
f (x) + g(x) = ( f + g)(x) D : D f I Dg
f (x) − g(x) = ( f − g)(x) D : D f I Dg

5. We can add, subtract, multiply, divide functions
f (x) + g(x) = ( f + g)(x) D : D f I Dg
f (x) − g(x) = ( f − g)(x) D : D f I Dg
f (x)⋅ g(x) = ( f ⋅ g)(x) D : D f I Dg

6. We can add, subtract, multiply, divide functions
f (x) + g(x) = ( f + g)(x) D : D f I Dg
f (x) − g(x) = ( f − g)(x) D : D f I Dg
f (x)⋅ g(x) = ( f ⋅ g)(x) D : D f I Dg
f (x) ⎛ f ⎞ D : D f I Dg
= ⎜ ⎟ ( x )
g(x) ⎝ g ⎠ and
g(x) ≠ 0

7. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2

8. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2
x−4 x−3
( f + g)(x) = + D: {x : x ≠ 1,2}
x −1 x − 2

9. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2
x−4 x−3
( f + g)(x) = + D: {x : x ≠ 1,2}
x −1 x − 2
x−4 x−3
( f − g)(x) = − D: {x : x ≠ 1,2}
x −1 x − 2

10. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2
x−4 x−3
( f + g)(x) = + D: {x : x ≠ 1,2}
x −1 x − 2
x−4 x−3
( f − g)(x) = − D: {x : x ≠ 1,2}
x −1 x − 2
⎛ x − 4 ⎞ ⎛ x − 3 ⎞
( f ⋅ g)(x) = ⎜
⎝ x − 1 ⎟ ⎜ x − 2 ⎟
⎠ ⎝ ⎠
D: {x : x ≠ 1,2}

11. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2
x−4 x−3
( f + g)(x) = + D: {x : x ≠ 1,2}
x −1 x − 2
x−4 x−3
( f − g)(x) = − D: {x : x ≠ 1,2}
x −1 x − 2
⎛ x − 4 ⎞ ⎛ x − 3 ⎞
( f ⋅ g)(x) = ⎜
⎝ x − 1 ⎟ ⎜ x − 2 ⎟
⎠ ⎝ ⎠
D: {x : x ≠ 1,2}
⎛ f ⎞ ⎛ x − 4 ⎞ ⎛ x − 2 ⎞
⎜ g ⎟ (x) = ⎜ x − 1 ⎟ ⎜ x − 3 ⎟
⎝ ⎠ ⎝ ⎠
D: {x : x ≠ 1,2, 3}
⎝ ⎠

12. x−4 x−3
Example: f (x) = g(x) =
x −1 x−2
x−4 x−3
( f + g)(x) = + D: {x : x ≠ 1,2}
x −1 x − 2
x−4 x−3
( f − g)(x) = − D: {x : x ≠ 1,2}
x −1 x − 2
⎛ x − 4 ⎞ ⎛ x − 3 ⎞
( f ⋅ g)(x) = ⎜
⎝ x − 1 ⎟ ⎜ x − 2 ⎟
⎠ ⎝ ⎠
D: {x : x ≠ 1,2}
⎛ f ⎞ ⎛ x − 4 ⎞ ⎛ x − 2 ⎞
⎜ g ⎟ (x) = ⎜ x − 1 ⎟ ⎜ x − 3 ⎟
⎝ ⎠ ⎝ ⎠
D: {x : x ≠ 1,2, 3}
⎝ ⎠
Be sure to read Example 2 in your textbook

13. Composite Functions

14. Composite Functions
The output of one function (the inner function)
is used as input for another function (the
outer function) notation : f ( g ( x ))

15. Composite Functions
The output of one function (the inner function)
is used as input for another function (the
outer function) notation : f ( g ( x ))
Example 1: y = x +1

16. Composite Functions
The output of one function (the inner function)
is used as input for another function (the
outer function) notation : f ( g ( x ))
Example 1: y = x +1
Inner function is done first x +1
Outer function is done second x

17. Composite Functions
The output of one function (the inner function)
is used as input for another function (the
outer function) notation : f ( g ( x ))
Example 1: y = x +1
Inner function is done first x +1
Outer function is done second x
g(x) = x + 1
f (x) = x
f (g(x)) = x + 1

18. Composite Functions
Example 2: y = sin ( 3θ )

19. Composite Functions
Example 2: y = sin ( 3θ )
Inner 3θ g(θ ) = 3θ

20. Composite Functions
Example 2: y = sin ( 3θ )
Inner 3θ g(θ ) = 3θ
Outer sin ( x ) f (x) = sin ( x )

21. Composite Functions
Example 2: y = sin ( 3θ )
Inner 3θ g(θ ) = 3θ
Outer sin ( x ) f (x) = sin ( x )
f (g(θ )) = sin ( 3θ )

22. Composite Functions
Example 3: y = sin 2 (θ )

23. Composite Functions
Example 3: y = sin 2 (θ )
Inner sin (θ ) g(θ ) = sin (θ )

24. Composite Functions
Example 3: y = sin 2 (θ )
Inner sin (θ ) g(θ ) = sin (θ )
Outer x 2
f (x) = x 2

25. Composite Functions
Example 3: y = sin 2 (θ )
Inner sin (θ ) g(θ ) = sin (θ )
Outer x 2
f (x) = x 2
f (g(θ )) = sin 2 (θ )

26. Composite Functions
Example 4: y = sin 2 ( 3θ )

27. Composite Functions
Example 4: y = sin 2 ( 3θ )
Innermost 3θ h(θ ) = 3θ

28. Composite Functions
Example 4: y = sin 2 ( 3θ )
Innermost 3θ h(θ ) = 3θ
Next Inner sin (θ ) g(θ ) = sin (θ )

29. Composite Functions
Example 4: y = sin 2 ( 3θ )
Innermost 3θ h(θ ) = 3θ
Next Inner sin (θ ) g(θ ) = sin (θ )
Outer x 2
f (x) = x 2

30. Composite Functions
Example 4: y = sin 2 ( 3θ )
Innermost 3θ h(θ ) = 3θ
Next Inner sin (θ ) g(θ ) = sin (θ )
Outer x 2
f (x) = x 2
f (g(h(θ ))) = sin 2 ( 3θ )

31. Composite Functions
Example 4: y = sin 2 ( 3θ )
Innermost 3θ h(θ ) = 3θ
Next Inner sin (θ ) g(θ ) = sin (θ )
Outer x 2
f (x) = x 2
f (g(h(θ ))) = sin 2 ( 3θ )
Be sure to read Example 7 in your textbook

32. Is f (g(x)) = g( f (x)) ?

33. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1
g(x) = x-3

34. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1
g(x) = x-3
f (g(x)) = 2(x − 3) + 1
= 2x − 6 + 1
= 2x − 5

35. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1
g(x) = x-3
f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3
= 2x − 6 + 1 = 2x − 2
= 2x − 5

36. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1
g(x) = x-3
f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3
= 2x − 6 + 1 = 2x − 2
= 2x − 5
Not the same

37. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1
g(x) = x-3
f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3
= 2x − 6 + 1 = 2x − 2
= 2x − 5
Not the same
Could they be the same?

38. Could f (g(x)) = g( f (x)) ?

39. Could f (g(x)) = g( f (x)) ?
x −1
Let f (x) = 2x + 1 and g(x) =
2

40. Could f (g(x)) = g( f (x)) ?
x −1
Let f (x) = 2x + 1 and g(x) =
2
⎛ x − 1 ⎞
f (g(x)) = 2 ⎜ ⎟ + 1
⎝ 2 ⎠
= x − 1+ 1
=x

41. Could f (g(x)) = g( f (x)) ?
x −1
Let f (x) = 2x + 1 and g(x) =
2
⎛ x − 1 ⎞ (2x + 1) − 1
f (g(x)) = 2 ⎜ ⎟ + 1 g( f (x)) =
⎝ 2 ⎠ 2
2x
= x − 1+ 1 =
2
=x =x

42. Could f (g(x)) = g( f (x)) ?
x −1
Let f (x) = 2x + 1 and g(x) =
2
⎛ x − 1 ⎞ (2x + 1) − 1
f (g(x)) = 2 ⎜ ⎟ + 1 g( f (x)) =
⎝ 2 ⎠ 2
2x
= x − 1+ 1 =
2
=x =x
These are the same. It happens when f(x)
and g(x) are inverses of each other.

43. HW #9
“There are precious few Einsteins among us.
Most brilliance arises from ordinary people
working together in extraordinary ways.”
Roger Von Oech