Functions!

What is a relation? (purple) Any set of ordered pairs: In a list In a table In a mapping diagram On a graph

What is a function? (purple) A one-to-one rule that gives one output for any input. ** x CANNOT repeat**

Ordered pairs (pink) We always write ORDERED PAIRS in curly braces with commas in between the ordered pairs. {(1,5), (-2,7), (4,3), (0,0) (-2,-3)} There are repeating x-values, so the example is NOT a function.

Input/output table (pink) The input (domain) is always “x”. They output (range) is always “y”. In a VERTICAL table, “x” always goes on the left side, & “y” on the right. In a HORIZONTAL table, “x” always goes on top, & “y” on the bottom. x y 2 3 -4 1 5 -2 x 1 7 3 y -4 9 -2 8 The first table is a function because none of the x-values repeat. The second table is NOT a function because there are repeating x-values (the 1’s).

Coordinate plane (Pink) The X-COORDINATE tells us to move left/right. The Y-COORDINATE tells us to move up/down. To check if a graph is a function, we use the VERTICAL LINE TEST. If you ran a vertical line over a graph and it ever touched more than one point, the graph is NOT a function.

Mapping diagram (pink) In the LEFT box/oval, write all input values (“x”). DO NOT REPEAT ANY! In the RIGHT box/oval, write all output values (“y”). DO NOT REPEAT ANY! Draw an arrow from each INPUT value to its OUTPUT value. Input Output The mapping to the left is NOT a function because there are multiple arrows coming from one input value. A function should have one arrow from each input value. 1 2 3 -2 1 7

Domain and Range (purple) Collection of inputs X-values Independent variables Range Collection of outputs Y-values Dependent variables

Independent and dependent Variables (Blue) Changing the independent variable Dependent variable Causes change in the X is always the independent variable. Y is always the dependent variable.

Function machine (purple) y=2x+3 x y 4 11 -1 1 2 7 21 3 5 9 13 2(4)+3

Function machines & function notation (yellow) A function can be though of as a machine that assigns the output to every input. f(x)=3x+4 Name of the function Tells what number to plug into the function f(x) means the output of the function f when the input is x. We often refer to this as y. THE FUNCTION!!

Let f(x)=x2+3 and g(x)=x+1 f(3) g(3) f(-2) g(0) f(2)+g(1) g(-1)+f(1) f(g(2)) g(f(0)) f(y) g(w)