Ch 5 Functions

1. Ch 5 Functions

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 4 –5 0 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, “afunctionis…a relation in which every input is paired with exactly one output”

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 samex-coordinates, it is NOT a function • Y-coordinates have no bearing in determining functions

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

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

11. –1 2 3 3 1 0 2 3 –2 0 2 –1 3 Example 5 Which mapping represents a function? Choice One Choice Two Choice 1 :40

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 replacexwith 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

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