FunctionsFunctionsFunctionsFunctions

ƒ(x) Function Notationsƒ(x) Function Notations

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.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. Can you determine if it is a relation or a function?

You can identify functions using tables or graphs. The graph below has more than

one output for each input. Is this 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 functionIf 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)

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.f(x) = yf(1) = 8

f(2) = 10f(3) = 12f(4) = 14

Linear FunctionsLinear FunctionsLinear FunctionsLinear Functions

Straight LinesStraight Lines

Writing equations of functionsUse 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 + b0 = m(2) – 40 = 2m – 4

0 + 4 = 2m -4 + 44 = 2m2 2m = 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 + b3 = m(1) + 1

3 = m + 13 – 1 = m + 1 – 1

m = 2

f(x) = 2x + 1

X Y

-2 -3

-1 -1

0 1

1 3

2 5

Physical ScienceThe 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

Temperature (°C) Temperature (°F) -10 14

0 32 25 77

50 122 100 212

Practice with Functions

1) 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)}. Which of the following values for x will make relation A a function?

a) 7b) 9c) 4

3) The following relation is a function.{(10, 12), (5, 3), (15, 10), (5, 6), (1, 0)}

• True

• False

4) Which of the relations below is a function?

a) {(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)}b) {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}c) {(0, 2), (0, 3), (0, 4), (0, 5), (0, 6)}

5) The graph of a relation is shown at the right. Is this relation a function?

a) yesb) noc) Cannot be

determine from a graph

6) Is the relation depicted in the table below a function?

a) yesb) noc) cannot be determined from a table

X 0 1 3 5 3 9

Y 8 9 10 6 10 7

7) The graph of a relation is shown below. Is the relation a function?a) yesb) noc) cannot be determined from a graph

8) Is the relation in the table below a function?

a) yesb) no

x -2 -1 0 1 2 3

y 5 5 5 5 5 5

9) The graph of a relation is shown below. Is the relation a function?a) yesb) noc) cannot be determined from a graph

10) The graph of a relation is shown below. Is the relation a function?

a) yesb) noc) cannot be determined from a

graph

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

a) 15b) 22c) 42

12) Which graph represents a function?