Post on 09-Jan-2016
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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