1. Ch 5 Functions Chapter 5: Functions
Section 5.1: Examples {1,2,9,12,14}
Section 5.4: Examples {2,3,4,5,6,7,9,10}

2. 5.1 Functions

3. Review
A relation between two variables x and y is a set of ordered pairs
An ordered pair consist of a x and y-coordinate
A relation may be viewed as ordered pairs, mapping design, table, equation, or written in sentences
x-values are inputs, domain, independent variable
y-values are outputs, range, dependent variable

4. Example 1 What is the domain? {0, 1, 2, 3, 4, 5} What is the range?
{-5, -4, -3, -2, -1, 0}

5. Example 2 What is the domain? {4, -5, 0, 9, -1} What is the range?–5 9 –1
Input –2 7
Output
What is the domain?
{4, -5, 0, 9, -1}
What is the range?
{-2, 7}

6. Is a relation a function?
What is a function?
According to the textbook, “a function is…a relation in which every input is paired with exactly one output” 6

7. Function, a special type of relation

8. Is a relation a function?
Focus on the x-coordinates, when given a relation
If the set of ordered pairs have different x-coordinates,
it IS A function
If the set of ordered pairs have same x-coordinates,
it is NOT a function
Y-coordinates have no bearing in determining functions

9. YES Example 3 Is this a function? Hint: Look only at the x-coordinates
:00
Is this a function?
Hint: Look only at the x-coordinates
YES

10. NO Example 4 Is this a function? Hint: Look only at the x-coordinates
:40
Example 4
Is this a function?
Hint: Look only at the x-coordinates NO

11. Choice 1 Example 5 Choice One Choice Two
:40
Example 5
Which mapping represents a function?
Choice One Choice Two 3 1 –1 2 2 –1 3 –2
Choice 1

12. Example 6
Which mapping represents a function?
A B. B

13. Function Notation f(x) means function of x and is read “f of x.”
f(x) = 2x + 1 is written in function notation.
The notation f(1) means to replace x with 1 resulting in the function value.
f(1) = 2x + 1
f(1) = 2(1) + 1
f(1) = 3

14. Functions Injections, Surjections and Bijections
Let f be a function from A to B.
Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique.
Note: this means that if a  b then f(a)  f(b).
Definition: f is onto or surjective if every y in B has a preimage.
Note: this means that for every y in B there must be an x in A such that f(x) = y.
Definition: f is bijective if it is surjective and injective (one-to-one and onto).

15. Functions A B a X b Y c Z d
Surjection but not an injection

16. Functions A B A B a a V V b b W W c c X X d d Y Y
Injection & a surjection, hence a bijection Z
Injection but not a surjection

17. Functions Inverse Functions
Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f-1, is the function from B to A defined as
f-1(y) = x iff f(x) = y

18. Functions Definition: Let S be a subset of B. Then
f-1(S) = {x | f(x)  S}
Note: f need not be a bijection for this definition to hold. Example: Let f be the following function: A B a X
f-1({Z}) = {c, d}
f-1({X, Y}) = {a, b} b Y c Z d

19. Functions Composition
Definition: Let f: B C, g: A B. The composition of f with g, denoted fg, is the function from A to C defined by
f  g(x) = f(g(x))

20. A g B f C a V h
Examples: b W i c X j d Y A
fg C a h b i c j d

21. Examples

22. Examples

23. Examples

24. Examples

25. Examples