Algebra 4.8 Functions and Relations

Functions and Relations A relation is a set of ordered pairs A function is a rule that establishes a relationship between two quantities, called the input and the output. Input = x The collection of the x values is known as the domain. Output = y. The collection of the y values is known as the range.

Function The most important part of the function: every input has only one output. An x cannot have two y’s. Examples: x y Is this relation a function? Yes. Each input has only one output. x y Is this relation a function? No. The input 1 has Two outputs 5 and 7.

A Birthday Example Imagine this relation: Input = All the students at CHSFA Output = The Birthday of that Student Is this a function?

Let’s See Person Birthday Jessica JonesMay 5th Bob BooneJune 21st Sally SmithNov. 18th Todd ThomasNov. 18th Is this a function? Does each input have only one output? Similarly, does each person have only one birthday? Since each input has only one output, then this is a function. Todd ThomasApril 3rd This would make this not a function because an input (Todd Thomas) would have more than one Output (Nov. 18 th and April 3 rd ). This cannot happen.

What if the input were the days of the year and the output was the people who had that day as their birthday? Let’s try as a class. Date People’s B-Day Jan. 1 st Stu Sanders Jan. 2 nd Lyn Lewis Jan. 3 rd John Jacobs Jan. 3 rd Sally Struthers Is this a function? Does each input have only one output? Similarly, does each date have only one person celebrating that day as their birthday? No. Since there is an input that has more than one output, then this is not a function.

Vertical Line Test A relation is a function if its graph passes the vertical line test. All vertical lines must intersect the graph only once. Are these functions? Do they pass the vertical line test? Yes! No! Yes!

Evaluate the Following Function in Function Notation f(x) = 2x – 3 when x = -2 f(-2) = 2(-2) – 3 = -4 – 3 = -7

You try! Evaluate the Following Function in Function Notation… f(x) = -7x – 3 when x = 4 f(4) = -7(4) – 3 = -28 – 3 = -31

Graphing a Function Graph f(x) = -1/2x + 4 Replace f(x) with y Graph y = -1/2x + 4 (0, 4)

Is the relation a function? If yes, state the domain and range. x y Yes, the relation is a function! The domain is 1, 2, 3, and 4 The range is -2, -3, -3, -5

Real-world functions Coke machine Toaster Lets come up with a few as a class.

Function Function- exactly one output for one input.

Function Rule Function Rule- equation that describes a functional relationship. X= total cokes 1= price per coke N= number of cokes needed

Writing a function rule # of loads 1234 Cost$2.75$5.50$8.25$11.00

# of loads 1234 Cost$2.75$5.50$8.25$11.00 Find the difference between each box

Put into fraction form and divide

Plug into equation

Check it... # of loads 1234 Cost$2.75$5.50$8.25$11.00

What if it doesn’t work? Time Cost of Bike Rental 1$10 2$16 3$22 4$28

What if it doesn’t work??? Time Cost of Bike Rental 1$10 2$16 3$22 4$28

What do I need to add?

The Rule is. Check it with another number. TimeCost 1$10 2$16 3$22 4$28