3.1 Functions and their Graphs

Relation – a mapping, or pairing of input values with output values.

Domain – set of input valuesRange – set of output values

Functions

Function – a relation is a function if there is exactly one output for each input.

Relations and Functions

Relations and functions between 2 quantities can be represented in many ways:

-mapping diagrams-tables-graphs-equations-verbal descriptions

Functions and Relations

Relations can be represented by ordered pairs (x, y) where x-coordinate is the 1st number and y-coordinate is the 2nd number.

Domain = First number (input)Range = Second number (output)

Functions and Relations

Quadrant IQuadrant II

Quadrant IVQuadrant III

y-axis

x-axis

Relations and Functions

Consider the following points:

{(a, 1), (b, 2), (c, 3), (e, 2)}List the domain: {a, b, c, e}List the range: {1, 2, 3}

Relations and Functions

Ex 2: {(3, 5), (4, -6), (2, -4), (-1, 5)}

List the Domain:{-1, 2, 3, 4}

List the Range: {-6, -4, 5}

Relations and Functions

How to tell if a relation is a function:-Only one output for each input (no x can be repeated)-Vertical Line Test: = no vertical line intersects the graph of the relation at more than 1 point.

Relations and Function

For the relation to be a function, no x may be repeated

Are the following Functions?1. {(1, 3), (-4, 2), (-6, 2), (0, 5)}

Yes = no x has been repeated2. {(1, 3), (-4, 2), (-6, 7), (1, 5)}

No = 1 was repeated

Relations and FunctionsInput Output

Age Weight16 220 Write as Ordered Pairs

17 125 (16, 220), (16, 122), (17, 179)

18 179 (18, 125), (18, 116)

116 Not a function!!

122

Relations and FunctionsInput OutputName Weight

Sue 125Mary 133Steve 159Carol 144Jose

Write as ordered pairs:{(Sue, 125), (Mary, 133), (Steve, 159), (Carol, 144), (Jose, 133)}

Yes, it is a function – no Input has been repeated

Relations and Functions

Vertical Line Test1. Is this a Function? Yes

Vertical Line Test, cont

Are the following functions?

No

Relations and Functions

Many functions can be represented by an equation in 2 variables:

Ex: y = 2x – 7 An ordered pair (x, y) of the equation is

a solution of the equation if the ordered pair is true when the values of x and y are substituted into it.

Relations and Functions

Ex: for the line y = 2x – 7, is the ordered pair (2, -3) a solution?

Substitute the values in for x and y

-3 = 2 (2) – 7-3 = -3 YES, the ordered pair is a

solution of the equation.

Relations and Functions

Are the following solutions to the equations?

1. y = 3x – 1 ; ((2, 5), (-2, -7)yes, yes

2. 2p + q = 5; (2, 3) (-5, 15)no, yes

Relations and Functions

In an equation, the input variable (x, domain) is the independent variable, and the output (y, range) is the dependent variable because it depends on the value of the input.

3.3 Functions - ContinuedFunction Notation – the symbol f(x) is read “f of x” and is used to notation a function.

Since a function is a relation, a function can be listed as a set of ordered pairs

(x, f(x)) where the domain is all values for which

the function is defined, and the range consists of the values of f(x) where x is the domain of f.

Functions, Cont

To determine the range of a given function (given the domain), simply plug the values in for the variable.Ex: f(x) = 3x + 2 Domain: {-1, 0, 5}

f(-1) = 3(-1) + 2 = -1f(0) = 3(0) + 2 = 2f(5) = 3(5) + 2 = 17

Functions, Cont

Find the range of f(x) = 2x – 7

given the D {-3, -1, 0, 7}

Functions, cont

Find the Domain of x:We assume the domain of a function to

be all real numbers that are an acceptable replacement for the variable (x).

To find the domain of a function, we must determine whether there are any unacceptable replacements.

Unacceptable Replacements

2 Things that make unacceptable replacements:

1. 0 in the denominator – if a value would make the denominator = 0, then the value is unacceptable.

2. (-) under the radical – if a value would cause the expression under the radical to be a negative number, then the value would be unacceptable.

Domain of a FUNCTION

Find the domain of:f(x) =

What happens if x = -3?f(-3) = = Undefined

Therefore the domain of the function is

D = {x| x 3}Which reads all x such that x does not equal -3

3

4

x

x

0

7

33

43

Domain of a function

Find the domain of the following functions:

1. 5.

2. 6.

3. 7.

4. 8.

x

xxf

2

23)(

72)( xxf

)6(

7)(

x

xxf

)2)(5()(

xx

xxf

xxf )(

8)( xxf

23)( xxf

43

4)(

x

xxf

Homework

p. 109 (13-20)

p. 114 (9, 11, 19)

p. 119 (1-27 odd)