© 2007 by S - Squared, Inc. All Rights Reserved.

Function: A relation where every input has exactly one output. Definition Note: A relation is any set of ordered pairs.

InputOutput This relation is a function because every input has exactly one output. This relation can be represented by the following ordered pairs: ( 2, 4 ) ( 5, 7 ) ( 1, 3 ) ( 7, 9 )

InputOutput This relation is NOT a function because inputs 1 and 2 each have two outputs. This relation can be represented by the following ordered pairs: ( 2, 4 )( 2, 7 ) ( 5, 7 ) ( 1, 3 )(1, 9 ) ( 7, 9 )

When a function is defined by an equation, identify the function by name, f(x), where x identifies the input and f is the name of the function. Note: A function name can be any letter, upper or lower case. Example:f(x) = x + 2 Read as: "f of x..." OR "f as a f unction of x …" Function Notation:

InputOutput f(x) = x + 2 This represents the function: xf(x) f(2) = f(5) = f(1) = f(7) = ( 2, 4 ) ( 5, 7 ) ( 1, 3 ) ( 7, 9 )

Let’s find a few ordered pairs for the function: f(x) = −2x + 4 (, ) f(0) = -2(0) + 4f(0) = 0 + 4f(0) = 4f(1) = -2(1) + 4f(1) = f(1) = 2f(-1) = -2(-1) + 4f(-1) = 2 + 4f(-1) = 6f(-.5) = -2(-.5) + 4f(-.5) = 1 + 4f(-.5) = 5f(.5) = -2(.5) + 4f(.5) = f(.5) = 3f(2) = -2(2) + 4f(2) = f(2) = Input Output Now, let’s see what happens if we graph these points!

(, ) Now, let’s see what happens if we graph these points! What happens if we keep graphing points? f(x) = −2x + 4 f(x) x

The function, f(x) = −2x + 4, is represented graphically as a line! f(x) x

We can use the graph to determine the “output” of the function. Find f (3)  Identify 3 on the x-axis 3  Locate the point on the graph where the x-coordinate is 3  Identify the y-coordinate for that point So, f (3) = − 2 f(x) x

Let’s check if this is correct! Remember our function? f(x) = −2x + 4 Find f (3) f(3) = − 2(3) + 4 = − = − 2 So, f (3) = − 2

Find f(4): Using the function definition: f(x) = −2x + 4 f(4) = −2(4) + 4 f(4) = −4

Find f(4): Using the graph: f(4) = −4 f(x) x