1. Math 160
Packet #1
Functions

2. function
input
output

3. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

4. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

5. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

6. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

7. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

8. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

9. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

10. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

11. Ex 1. Ex 2. Find 𝑓(2) if 𝑓 𝑥 = 𝑥 2 +3.
𝒙 𝟐 +𝟑

12. Ex 3. Suppose 𝑓 𝑥 = 𝑥+12 9−𝑥 . Find 𝑓(10).

13. Ex 3. Suppose 𝑓 𝑥 = 𝑥+12 9−𝑥 . Find 𝑓(10).

14. Ex 3. Suppose 𝑓 𝑥 = 𝑥+12 9−𝑥 . Find 𝑓(10).

15. Ex 3. Suppose 𝑓 𝑥 = 𝑥+12 9−𝑥 . Find 𝑓(10).

16. Ex 4. Suppose 𝑓 𝑥 = 𝑥 3 + 𝑥 2𝑥−1 . Find 𝑓 .

17. Ex 4. Suppose 𝑓 𝑥 = 𝑥 3 + 𝑥 2𝑥−1 . Find 𝑓 .

18. Ex 5. Suppose 𝑓 𝑥 =2 𝑥 2 −3𝑥. Find 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ (called the difference quotient) and simplify by canceling the factor of ℎ.

19. Ex 6. Suppose 𝑓 𝑥 = 3 𝑥 . Find 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ and simplify by canceling the factor of ℎ.

20. Ex 7. Suppose 𝑓 𝑥 = 𝑥−5 . Find 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ and simplify by canceling the factor of ℎ.

21. Sometimes functions have different formulas depending on your input
Sometimes functions have different formulas depending on your input. These are called piecewise-defined functions. Ex 6. Evaluate the piecewise defined function at the indicated values. 𝑓 𝑥 = 2− 𝑥 2 3𝑥−3 −2 if 𝑥≤−1 if−1<𝑥≤4 if 𝑥>4 𝑓 −2 = 𝑓 −1 = 𝑓 0 = 𝑓 4 = 𝑓 5 = 𝑓 =

22. Sometimes functions have different formulas depending on your input
Sometimes functions have different formulas depending on your input. These are called piecewise-defined functions. Ex 8. Evaluate the piecewise-defined function at the indicated values. 𝑓 𝑥 = 2− 𝑥 2 3𝑥−3 −2 if 𝑥<−1 if−1<𝑥≤4 if 𝑥>4 𝑓 −2 = 𝑓 −1 = 𝑓 0 = 𝑓 4 = 𝑓 5 = 𝑓 =

23. The __________ of a function is the set of all possible ____________.

24. The __________ of a function is the set of all possible ____________.
domain

25. The __________ of a function is the set of all possible ____________.
domain
inputs

26. The __________ of a function is the set of all possible ____________.
domain
inputs
range

27. The __________ of a function is the set of all possible ____________.
domain
inputs
range
outputs

28. Domain
≠0

29. Domain
≠0 
≥0

30. Domain
≠0 
≥0
log 
>0

31. Ex 9. Find the domain of each of the following functions

32. Suppose you have two functions, 𝑓(𝑥) and 𝑔(𝑥)
Suppose you have two functions, 𝑓(𝑥) and 𝑔(𝑥). You can add, subtract, multiply, and divide 𝑓 and 𝑔 to get new functions: 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔 𝑥 𝑓𝑔 𝑥 =𝑓 𝑥 𝑔(𝑥) 𝑓 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥

33. Suppose you have two functions, 𝑓(𝑥) and 𝑔(𝑥)
Suppose you have two functions, 𝑓(𝑥) and 𝑔(𝑥). You can add, subtract, multiply, and divide 𝑓 and 𝑔 to get new functions: 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔 𝑥 𝑓𝑔 𝑥 =𝑓 𝑥 𝑔(𝑥) 𝑓 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥

34. ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥
ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥 . Then… 𝑓+𝑔 𝑥 = 1 𝑥−2 + 𝑥 𝑓 𝑔 𝑥 = 1 𝑥−2 𝑥 = 1 𝑥−2 𝑥

35. ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥
ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥 . Then… 𝑓+𝑔 𝑥 = 1 𝑥−2 + 𝑥 𝑓 𝑔 𝑥 = 1 𝑥−2 𝑥 = 1 𝑥−2 𝑥

36. ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥
ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥 . Then… 𝑓+𝑔 𝑥 = 1 𝑥−2 + 𝑥 𝑓 𝑔 𝑥 = 1 𝑥−2 𝑥 = 1 𝑥−2 𝑥

37. ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥
ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥 . Then… 𝑓+𝑔 𝑥 = 1 𝑥−2 + 𝑥 𝑓 𝑔 𝑥 = 1 𝑥−2 𝑥 = 1 𝑥−2 𝑥

38. ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥
ex: Suppose 𝑓 𝑥 = 1 𝑥−2 and 𝑔 𝑥 = 𝑥 . Then… 𝑓+𝑔 𝑥 = 1 𝑥−2 + 𝑥 𝑓 𝑔 𝑥 = 1 𝑥−2 𝑥 = 1 𝑥−2 𝑥

39. The domains of 𝑓+𝑔, 𝑓−𝑔, and 𝑓𝑔 are all the numbers that can be plugged into both 𝑓(𝑥) and 𝑔(𝑥) (that is, it’s the intersection of the domains of 𝑓 and 𝑔). For 𝑓/𝑔 it’s the same, though you also have to take out any numbers that make 𝑔 𝑥 =0.

40. The domains of 𝑓+𝑔, 𝑓−𝑔, and 𝑓𝑔 are all the numbers that can be plugged into both 𝑓(𝑥) and 𝑔(𝑥) (that is, it’s the intersection of the domains of 𝑓 and 𝑔). For 𝑓/𝑔 it’s the same, though you also have to take out any numbers that make 𝑔 𝑥 =0.