1. Section 7.1 Introduction to Function

2. Relation A connection between two things, input and output. Examples of relations are school and study family and siblings sports and hockey As you can see there are many words that can describe the first word. What ideas do you have?

3. Formal Definition for Relation Relation – A connection between two things – A pairing of elements of one set with elements of a second set. The one set and second set have proper algebra terms. – One set is referred to as the domain. – Second set is referred to as the range. – A set of ordered pairs

4. Example: Which are Relations? A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 5 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

5. Which are Relations They are ALL Relations

6. Special types of Relations Function One – to – One Function These are special because there is a unique connection between the input (domain) and the output (range) We will have different definitions for function and one – to – one function

7. Function A relation in which each element of the domain is paired to exactly one element in the range Given an equation you COULD see f(x) Given a graph a function will pass the Vertical Line Test (VLT) – Vertical Line Test = if you can draw vertical lines on your graph and only pass through one point on the graph then your graph is a function.

8. Example: Which are Functions? A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 5 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

9. Example: Which are Functions? A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} F B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F F(x) = 2x – 5 F Y = x² + 5graph the equation F RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

10. One – to – One Function A function in which each element of the range is paired to exactly one element in the domain. No special indication for an equation. Must pass the Vertical Line Test AND the Horizontal Line Test

11. Example: Which are 1-1 Functions? A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 5 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

12. Example: Which are 1-1 Functions? A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} 1-1 B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 51-1 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

13. Domain / Range Domain – Given an ordered pair (x, y) the first element, the input, the x element, the values on the horizontal line are all referred to as the domain. Range – Given an ordered pair (x, y) the second element, the output, the y element, the values on the vertical line are all referred to as the range.

14. Example: Define the Domain A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 5 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

15. Example: Define the Range A = {(-2, -9), (-1, -7), (0, -5), (1, -3), (2, -1)} B = {(1, 5), (-2, 3/2), (0, 2), (-2, 4), (-1/5, 0)} C = {(-1, 2), (1/3, -2), (5, -2), (1/3, 2), (0, -1/9)} F(x) = 2x – 5 Y = x² + 5 RedBakersfield condors GreenMoney GoldCal Poly BlueBakersfield College WhiteUSA Flag

16. Evaluating a Function Evaluating a function either has you – Substitute values Given a function and a value. Substitute value in for proper variable. Solve for other variable. – Play with a graph Given a graph and a value. Find that value on the correct axis. Find the connection of that value and the given graph Connect that point to the other axis. Find the new value.

17. Evaluating a function Given f(x) = x² - 2x + 3 Evaluate – f(-3) f(-3) = (-3)² - 2(-3) + 3 f(-3) = 9 + 6 + 3 =18 – f(2) – f(0) – f(-5) – f(1/2)

18. Evaluating a function Given H(t) = t – 12 evaluate – H(y) – H(y + 2) – H(2y) – H(y) +2

19. Evaluating a function Given the graph evaluate f(1) f(2) f(4) f(1/2)

20. Evaluate the function Given the graph evaluate f(1) f(0) Find x if f(x) = 5 f(x) = 0 f(x) = -10 f(x) = 20

21. Homework 7.1 – # 9, 16, 27, 31, 33, 37, 38, 40, 43, 46, 47, 57