Functions vs. Relations

1. Functions vs. Relations Lesson 1.1

2. Definitions • Relations – how variables influence one another • Functions – a relationship between variables in which each input has one and only one output. A function is always a relation, but a relation is not always a function. • Domain – the set of inputs or x-values that can be evaluated for a particular function • Range – the set of outputs or y-values that can be obtained for a particular function

3. What is a function? • A function is defined as a relation where there is only one output for any given input. • Example from our class: your birthday doesn’t change. No matter how often or at what time of day you ask your classmate for their birthday, the answer will stay the same. • What are other examples of functions?

4. Examples of functions • Functions can be found in a number of ways from coordinates to tables to graphs. • What are you looking for to define a function? Every input has a distinct output. • Graphically: Tabular: Coordinates: (2, 4), (5, 5), (-2, 6), (-7, 7)

5. Functions vs. Relations • Function • Ordered Pairs (1, 5), (2, 7), (3, 6), (2, 7) Each input has only one associated output • Relation • Ordered Pairs (1, 5), (2, 7), (3, 6), (2, 9) At least one input has multiple outputs

6. Functions vs. Relations • Graph Passes Vertical Line Test • Graph Fails Vertical Line Test

7. Functions vs. Relations • Input/Output chart Each input has only one output. • Input/Output chart At least one input has more than one output

8. Functions vs. Relations • Functions • x-y tables Each x has only one y • Relations • x-y tables At least one x has more than one y

9. Function or not? • The amount of money that the ticket taker at AMC makes in relation to the number of hours he works. • The amount of money that a waitress at Chili’s makes in relation to the number of hours she works. • The amount of money that a Chick-fil-A employee makes in relation to the number of hours worked.

10. Function or not? • The heights of students in the class in relation to their student ID numbers. • The heights of students in the class in relation to their shoe size. • The heights of students in the class in relation to their birthday.

11. Function Notation • y = x + 8 – You’ve seen this before! • Remember these tables: • Now, we simply replace y with f(x) where x is our input, and f(x) is the output that we’re trying to find.

12. Function Notation • We can use any of the formulas we know for function notation. (Let’s try to stick with formulas that only have one variable) • Example: the formula for the area of a circle is A = πr2 • In function notation, the formula becomes Area(r)= πr2 or we can abbreviate it to A(r) = πr2 • What is our input in this function? What is our output? • What are a couple other formulas we know that could be written in function notation?

13. Evaluating a function • To evaluate a function in function notation, we simply replace the variable with the value. • For example, the function f(x) = x + 5 when x = 4 • We write that f(4) = (4) + 5 = 9 • WE SIMPLY REPLACED THE “X” WITH ITS VALUE.

14. You Try • Evaluate the Following Functions • F(x) = 3x + 6 G(x) = (4 – x) – 2 • 1. F(5) 2.F(-3) • 3. G(5) 4.G(-3)

15. Algebraic Functions • F(x) = x + 5 • F(x) = -3x + 2 • F(x) = 1/x • F(x) = √x • Will any of the above expressions result in two answers for a given x?