Section 1.3 Functions•Objective:

•To evaluate functions and find their domains•To evaluate difference quotients.

Assignment

p. 141#1, 5, 6, 15, 18, 31, 34, 37

This is pretty functional

Function = set of ordered pairs◦A mapping from one set to another

(-2, 4)

(2, 4)

(1, 1)

(-1, 1)

(0, 0)

x y

-2

2

1

-1

0

4

4

1

1

0

What is this function?

What is NOT a function?

Values from the independent variable (x) cannot be mapped to two or more dependent variables (y)

(0, 1)(0, -2)

Fails, think about the vertical line test.

Which are functions?

Easternconference

BullsHeatKnicksMagic

Name that Function

Input 0 3 9 12 15

Output 3 3 3 3 3

Input 0 1 1 2 2

Output 4 5 6 7 8

Input -1 0 1 3 5

Output 3 5 7 11 15

Solve Algebraically

Given an equation

Solve for yDetermine if each x value will have at

most one y value.

41 2 xxy 22 yx

Solve Algebraically

52 xxy xy 2

Function Notation

Functions, functions, functions!!!

What do we know:◦“y =“ typically can be made into a graph

Our change:◦Think about a graph not in terms of x -> y, but

instead as x -> f(x)◦Since we say that y is a function of x , we can

replace y with f(x) making it “f(x) =“

1

2

3

2

4

6

X f Y

1

2

3

X f f(x)

2

4

6

If f(x) = x2 – 4x , find the following.a)

b)

c) f(x+ h) =

Remember: ◦y = f(x). ◦ f is the name of the function, not another variable.

f (3) 32 439 12

3

f (a) a2 4a

Domain of a function

p. 143# 51, 54, 55, 63, 65, 71, 74

Domain of a function:◦all x such that the matching y is a real number.

◦consider only three situations Polynomials domain is (-, ).

◦3x, , Fractions cannot have any number in the domain

that makes the denominator zero.

Radicals if the index is even, then the radicand must be nonnegative.

xx 2 743 xx

32

32 x

x

2

23 2

x

x5

12 x

24 x 13 x 4 x

Domain of a function: Polynomials domain is (-, ).

◦ f(x) = x3 + 3x + 1◦This is a polynomial, so the domain is (-, ).

Fractions cannot have any number in the domain that makes the denominator zero since that would make it undefined.

◦x + 2 0, so the domain is (-, -2) (-2, )

Radicals if the index is even, then the radicand must be nonnegative.

5 – x 0, so the domain is (–, 5]

f ( x) 2

x 2

f ( x) 5 x4

What is the domain of each function?

x

x

2

3

x8

214

22

xx

x

(- ∞, -3) U (-3, 7) U (7, ∞)

(- ∞, 8]

[-3, 2) U (2, ∞)

Difference Quotient

h

xfhxf )()( Finds the slope of a line between two points on a function

x tells you where you are looking at the function because different parts will have different slopes

h = distance between the two points

http://media.lanecc.edu/users/gettyst/Geogebra/difference_quotient.html

It’s time to teach you Calculus

If f ( x) x2 x 1, find f (x h ) f (x)

h

f ( x h) f (x)h

x h 2 x h 1 x 2 x 1

h

x2 2xh h2 x h 1 x2 x 1

h

2xh h2 hh

2x h 1

Application

Assignment

Pre-read section 1.4◦On a sheet of loose-leaf paper, take notes on

what you believe is important to the section◦Try to think about what material will be

important to the objectives in the book◦Will be an assessment for me to see how well

you read a textbook to see what we can work on together