ƒ(x) Function Notations Functions ƒ(x) Function Notations

A relation is a pairing between two sets A relation is a pairing between two sets. A function is a relation in which each x-value has only one y-value Functions can be represented in many ways including tables, graphs and equations. Take for example the equation y = 2x - 3. This equation has an important characteristic. For each value of x, you find exactly one value of y.

Relation: A relation is simply a set of ordered pairs Relation: A relation is simply a set of ordered pairs. A relation can be any set of ordered pairs. No special rules need apply. The following is an example of a relation: {(1,2)(2,4)(3,5)(2,6)(1,-3)} The graph at the right shows that a vertical line may intersect more than one point in a relation.

Function: A function is a set of ordered pairs in which each x-value has only ONE y-value associated with it. The relation we just discussed {(1,2),(2,4,)(3,5)(2,6)(1,-3)} is NOT a function because the x-value 2 is paired with a y-value of 4 and 6. Similarly, the x-value of 1 is paired with the y-value of 2 and -3

The previous relation can be altered to become a function by removing the ordered pairs where the x-value is used twice. Function: {(1,2)(2,4)(3,5)} The graph at the left shows that a vertical line intersects only ONE point in a function. This is called the vertical line test for functions.

Function – an input-output relationship that has exactly one output for each input. Domain – the set of all input (x)values of a function. Range – The set of all output (y)values in a function. Function notation – the notation used to describe a function. Example f(x) is read “f of x.” f(1) is read “f of 1.” Linear function – a function whose graph is a straight line.

To determine if a relationship is a function, verify that each input has exactly one output. Using tables is one way to verify functions.

Look at the function below Look at the function below. Can you determine if it is a relation or a function?

You can identify functions using tables or graphs You can identify functions using tables or graphs. The graph below has more than one output for each input. Is this a function?

Determine whether the order pairs make a function. A relation can be represented by a set of order pairs (x, y) . The first number, x, is a member of the domain and the second number, y, is a member of the range. Determine whether the order pairs make a function. {(-1, 7), (0, 3), (1, 5), (0, -3)} {(0, 2), (2, 4), (4, 8), (8, 10)}

Vertical-Line Test for a function If no vertical line in the coordinate plane intersects a graph in more than one point, then the graph represents a function. (You can use a pencil held vertically to test)

For the function y = 2x - 1, find f(0), f(2), and f(-1). Evaluating Functions For the function y = 2x - 1, find f(0), f(2), and f(-1). y = 2x – 1 f(x) = 2x -1 Write in function notation. f(0) = 2(0) – 1 = -1 f(2) = 2(2) – 1 = 3 f(-1) = 2(-1) – 1 = -3

Find f(1), f(2), f(3), and f(4). Read the graph to find y for each x Find f(1), f(2), f(3), and f(4). Read the graph to find y for each x. f(x) = y f(1) = 8 f(2) = 10 f(3) = 12 f(4) = 14

Linear Functions Straight Lines

Writing equations of functions Use the equation f(x) = mx + b Writing equations of functions Use the equation f(x) = mx + b. Find b (y-intercept) = -4 Locate a point on the line, such as (2, 0). Substitute the values into your equation. f(x) = mx + b 0 = m(2) – 4 0 = 2m – 4 0 + 4 = 2m -4 + 4 4 = 2m 2 2 m = 2 f(x) = 2x - 4

Writing an equation using a table The y-intercept can be identified from the table, (0, 1) Pick a point, (1, 3) and substitute your point and y-intercept into your equation. f(x) = mx + b 3 = m(1) + 1 3 = m + 1 3 – 1 = m + 1 – 1 m = 2 f(x) = 2x + 1

Physical Science The relationship between the two temperatures in the table are linear. Write a rule for Fahrenheit temperature as a function of Celsius temperature. f(x) = mx + b, where x is Celsius and y is Fahrenheit

Practice with Functions Which of the relations below is a function? a) {(2, 3), (3, 4), (5, 1), (2, 4)} b) {(2, 3), (3, 4), (6, 2), (7, 3)} c) {(2, 3), ( 3, 4), (6, 2), (3, 3)}

2) Given the relation A = {(5, 2), (7, 4), (9, 10), (x, 5)} 2) Given the relation A = {(5, 2), (7, 4), (9, 10), (x, 5)}. Which of the following values for x will make relation A a function? a) 7 b) 9 c) 4

The following relation is a function True False

4) Which of the relations below is a function? {(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)} {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)} {(0, 2), (0, 3), (0, 4), (0, 5), (0, 6)}

The graph of a relation is shown at the right The graph of a relation is shown at the right. Is this relation a function? a) yes b) no c) Cannot be determine from a graph

6) Is the relation depicted in the table below a function 6) Is the relation depicted in the table below a function? a) yes b) no c) cannot be determined from a table

The graph of a relation is shown below. Is the relation a function The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph

8) Is the relation in the table below a function? a) yes b) no

The graph of a relation is shown below. Is the relation a function The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph

10) The graph of a relation is shown below. Is the relation a function? a) yes b) no c) cannot be determined from a graph

11) Given f(x) = 3x + 7, find f(5). 15 22 42

Which graph represents a function?