1. 6-5 operations with functions
Chapter 6
6-5 operations with functions

2. Objectives Add, subtract, multiply, and divide functions.
Write and evaluate composite functions.

3. Operations with functions
You can perform operations on functions in much the same way that you perform operations on numbers or expressions. You can add, subtract, multiply, or divide functions by operating on their rules.

4. Notation

5. Example 1A: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.
(f + g)(x)
Solution:
(f + g)(x) = f(x) + g(x)
= (4x2 + 3x – 1) + (6x + 2)
= 4x2 + 9x + 1

6. Example 1B: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.
Solution:
(f – g)(x) = f(x) – g(x)
= (4x2 + 3x – 1) – (6x + 2)
= 4x2 + 3x – 1 – 6x – 2
(f – g)(x)
= 4x2 – 3x – 3

7. Check It Out! Example 1a
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.
(f + g)(x)
= x2
(f – g)(x)
= –x2 + 10x – 12

8. When you divide functions, be sure to note any domain restrictions that may arise.

9. Example 2A: Multiplying and Dividing Functions
Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function.
(fg)(x)
Solution
(fg)(x) = f(x) ● g(x)
= (6x2 – x – 12) (2x – 3)
6x2 (2x – 3) – x(2x – 3) – 12(2x – 3)
12x3 – 18x2 – 2x2 + 3x – 24x + 36
= 12x3 – 20x2 – 21x + 36

10. Example 2B: Multiplying and Dividing Functions
Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function
( f/g )(x)

11. ( ) Solution f f(x) (x) = g g(x) 6x2 – x –12 = 2x – 3 (2x – 3)(3x + 4)
( )
(x) f g
f(x)
g(x) =
6x2 – x –12
2x – 3 = =
(2x – 3)(3x + 4)
2x – 3 =
(2x – 3)(3x +4)
(2x – 3)
3x + 4, where x ≠ 3/2

12. Check it out
Given f(x) = x + 2 and g(x) = x2 – 4, find each function.
(fg)(x)
= x – 2, where x ≠ –2
= x3 + 2x2 – 4x – 8
( )
(x) g f

13. Composition of functions
Another function operation uses the output from one function as the input for a second function. This operation is called the composition of functions.

14. Composition of function
The order of function operations is the same as the order of operations for numbers and expressions. To find f(g(3)), evaluate g(3) first and then substitute the result into f.

15. Example 3A: Evaluating Composite Functions
Given f(x) = 2x and g(x) = 7 – x, find each value.
Solution:
Step 1 Find g(4)
f(g(4))
So f(g(4)) = 8.
g(4) = 7 – 4
= 3
Step 2 Find f(3)
f(3) = 23
= 8

16. Example
Given f(x) = 2x and g(x) = 7 – x, find each value.
g(f(4))

17. Check It Out! Example 3a
Given f(x) = 2x – 3 and g(x) = x2, find each value.
f(g(3))
So f(g(3)) = 15.
g(f(3))
So g(f(3)) = 9

18. Operations with functions
You can use algebraic expressions as well as numbers as inputs into functions. To find a rule for f(g(x)), substitute the rule for g into f.

19. Example 4A: Writing Composite Functions
Given f(x) = x2 – 1 and g(x) = 𝒙/(𝟏−𝒙) , write each composite function. State the domain of each.
f(g(x))
Solution
f(g(x)) = f( x
1 – x
–1 + 2x
(1 – x)2 =
= ( )2 – 1 x
1 – x
The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because g(1) is undefined.

20. Example
Given f(x) = x2 – 1 and g(x) = 𝒙 𝟏−𝒙 , write each composite function. State the domain of each.
g(f(x))
The domain of g(f(x)) is x ≠ or {x|x ≠ } because f( ) = 1 and g(1) is undefined.

21. Check It Out! Example 4a
Given f(x) = 3x – 4 and g(x) = , write each composite. State the domain of each.
f(g(x))
The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.
g(f(x))

22. Composite functions
Composite functions can be used to simplify a series of functions.

23. Application
Jake imports furniture from Mexico. The exchange rate is pesos per U.S. dollar. The cost of each piece of furniture is given in pesos. The total cost of each piece of furniture includes a 15% service charge.
A. Write a composite function to represent the total cost of a piece of furniture in dollars if the cost of the item is c pesos.

24. Solution Step 1 Write a function for the total cost in U.S. dollars.
P(c) = c c
= 1.15c
Step 2 Write a function for the cost in dollars based on the cost in pesos.
D(c) = c
11.30

25. Solutiob Step 3 Find the composition D(P(c)). 11.30 1800 11.30 c
= 1.15 ( ) c
11.30
B. Find the total cost of a table in dollars if it costs 1800 pesos.
Evaluate the composite function for c = 1800.
D(P(c) ) = 1.15 ( )
1800
11.30 ≈

26. Student guided practice
Do even numbers from 2-14

27. Homework
Do even numbers in your book page 446