functions: f(x)= means y= a relation is a function if every x value is paired with one y value (no...
TRANSCRIPT
Functions: f(x)= means y=Functions: f(x)= means y=
A relation is a function if every x value is paired with one y value (no repeat x’s)
A relation is a function if it passes the verticalline test-any vertical line drawn will pass through the function in at most one point.
Function notationFunction notation
f(x) = x + 8Find f(2)This means “let x = 2”
f(2) = 2 + 8= 10
Function notationFunction notation
f(x) = 2x - 18Find f(-1)Find f(2a)Find f(b+3)
Function notationFunction notation
f(x) = 2x - 18Find f(-1)= -20Find f(2a) = 2(2a)-18=4a-18Find f(b+3)=2(b+3)-18=2b+6-18=2b-12
Find: f(-1) and f(3)Find: f(-1) and f(3)
f(x)
f(-1)=2 and f(3)=2f(-1)=2 and f(3)=2
Ordered pair examples Ordered pair examples
S={(-3,5),(4,7),(0,5)} S is a functionP={(5,-3),(7,4),(5,0)} P is not a function
Is F a function?
F = {(-3,7),(4,7),(-3,5)}
Graphs-number 4 fails –not a Graphs-number 4 fails –not a functionfunction
Which is not a function?Which is not a function?
equationsequations
f (x)=x2 −1
f(x) = x−1
x2 + y2 =25x=7y=5
In general, x=c are not functions, and relationsthat have a “y2” are not functions
Which are functions?
answersanswers
x =7
x2 + y2 =25
All but these two are functions
This is a circle
This is a vertical line
One to one and ontoOne to one and onto
One to one functions pass the horizontal line test and have no repeat y values
Onto means all y values are used
235
478
One to one and onto
-279
360
Not one to one or onto
D R D R
Graphs of one to oneGraphs of one to one
Which is not a function?Which are not one to one?
Equations are they one to Equations are they one to one?one?
y=5x
f(x) = x + 3
y= x+ 4
y=x2 +2
Use the graphing calculator to see if these are one to one functions:
Use it to see that this function is not One to one:
Domain (x) and Range (y)Domain (x) and Range (y)
Domain: what x values can be or can’t be
Range: what y values the function has.
Domain: left to right Range: bottom to top
Notations: both mean all reals means all reals except 2
x ∈ R , (−∞,∞)x ∈ R\ {2}
Is this a one to one function?Is this a one to one function?
7
8
9
7
8
9
4
5
6
4
5
6
domain Range
Make a list of ordered pairs….
List all x’s and list all y’sList all x’s and list all y’s
7
8
9
7
8
9
4
5
6
4
5
6
domain Range
examplesexamples
S = {(-3,5),(4,7),(0,5)}
Domain = {-3,4,0}Range = {5,7}
Restricted domains-denominators Restricted domains-denominators and radicals!and radicals!
y=1
x−9
f(x) = x−2
f(x) =xx+1
Set denominator =0 and solve
Set and solve
Set x + 1>0 and solve
x−2 ≥0
Answers to domainsAnswers to domains
y=1
x−9
f(x) = x−2
f(x) =xx+1
X - 9=0X=9 D:
x + 1>0 D: x >-1
x−2 ≥0x≥2
x ∈ R\ {9}
D: [2,∞)
(−1,∞)
RangeRange
Look at the graph from the bottom up And state the y values:
Y=x2 +2
The lowest pointIs (0,2)
RangeRange
Look at the graph from the bottom up And state the y values:
Y=x2 +2
Range:
[2,∞)
assessmentassessment
1) f (x)=x2 −4
2) f(x) = x−3 +2
State the domain and range of the followingFunctions:
Domain for both is all reals!Domain for both is all reals!
1) f (x)=x2 −4
2) f(x) = x−3 +2
State the domain and range of the followingFunctions:
R :[−4,∞)
R :[2,∞)
In interval notationIn interval notation
Find the domain and range:
answersanswers
Find the domain and range:
Domain: [-5,4]Range: [-4,3]
answersanswers
D:
R:
answersanswers
(−3,4)[−4,5)
D:
R:
compositionscompositions
fοg(x) or f(g(x))A function inside a function-above: the g functionIs placed inside the f function.
f (x)=x+1 and g(x) =x2
find f(g(3))
Find g(3)Take that answer and findF(that answer)
compositionscompositions
fοg(x) or f(g(x))A function inside a function-above: the g functionIs placed inside the f function.
f (x)=x+1 and g(x) =x2
f(g(3)) =10
g(3)=9Take that answer and findf(9)=9+1=10
compositionscompositions
fοg(x) or f(g(x))A function inside a function-above: the g functionIs placed inside the f function.
Steps:1)write f: x+ 12)Replace x: ( ) + 13)Put g in ( ): (x2) + 14)Simplify: x2 + 1
examplesexamples
given : f (x)=x2 +2x and g(x) =3x
find : f(g(4))fοg(x)gο f(x)
Examples-answersExamples-answers
given : f (x)=x2 +2x and g(x) =3x
find : f(g(4)) =3(4) =12 then 122 +2(12) =168
fοg(x) =(3x)2 +2(3x) =9x2 +6x
gο f(x) =3(x2 +2x) =3x2 +6x
Do and hand inDo and hand in
f (x)=x2 −4
g(x) =2x
findf og(3)
f og(x)
g( f(x))
Given:
Inverses f Inverses f -1-1 (x) (x)
If a function is one to one then it is an inversefunction. “Swap” x’s and y’s. The graph of afunction and it’s inverse are a reflection in y=x
Example:
F = {(1,-2), (4, -8), (5, -10)}
Find the inverse….F -1 = {(-2,1), (-8,4), (-10,5)}
Finding an inverse function.Finding an inverse function.
Given f(x) = 2x + 4, find the inverse…“swap and solve”:
y=2x+ 4X =2Y + 4X−4 =2YX−4
2=
2Y2
Y =X2
−2
f−1(x) =12
X−2
Y=SWAP X & Y
SOLVE FOR Y
Simplify
Change y to f -1
examplesexamples
Find the inverse:
examplesexamples
f (x)=x3 −1
x=y3 −1
x+1=y3
x+13 =y so f−1(x) = x+13
f (x) =3x−9x=3y−9x+9 =3y
y=x+93
,so f−1(x) =13
x+ 3 Y=x
Y=1/3x+3
Y=3x-9
Example 3Example 3
f (x)=x+2
x
x=y+2
yxy=y+2xy−y=2y(x−1)=2y(x−1)(x−1)
=2
(x−1)
y=2
x−1, so ... f−1(x) =
2x−1
Swap x & y
Cross multiply
Get y’s together
Factor out y
Solve for y
f(f f(f -1-1 (x))=x (x))=x
We call the identity function y=x
When we compose inverses of each other, ineither order, we get x as an answer. Example:
f (x)=3x−9
f−1(x) =13
x+ 3
f of−1(x) =3(13
x+ 3)−9
=x+9 −9=x
Distribute the 3
Restricting domainsRestricting domains
When a function is not one to one, we can find a solution if we restrict the domain
y=x2 +2 , x≥0
So now we are only consideringhalf of our graph. and can find the inverse.
solutionsolution
y=x2 +2 , x≥0
x=y2 +2
x−2 =y2
y= x−2
f−1 = x−2Y=x
practicepractice
f (x)=2x −4
g(x) =x2 + 4, x≥0
find
f−1(x)
g−1(x)
f−1 of (x)
Given:
Transformations of functionsTransformations of functions
Vertical shifts:F(x) + a is a shift up a unitsF(x) – a is a shift down a units
Horizontal shifts:x+ a is a shift to the left a unitsx – a is a shift to the right a units
So x is confused and y is not!
Library of functionsLibrary of functions
Y = x2
y= x
y=x3 y= x
examplesexamples
f (x)= x−8
f(x) =(x+1)2
f(x) = x −6
f(x) =(x+5)2 −2
Describe the shifts of the functions: x or the x2
examplesexamples
f (x)= x−8
f(x) =(x+1)2
f(x) = x −6
f(x) =(x+5)2 +2
Describe the shifts of the functions: x or the x2
Horizontal right 8
Horizontal left 1
Vertical down 6
Horizontal left 5 and vertical up 2
reflectionsreflections
A reflection in the y axis is a negation of xA reflection in the x axis is a negation of y (or f(x))
Examples:
f (x)= x+ 4
f(x) =(x−7)2
f (x)=−x+ 4
f(x) =(−x−7)2
f (x)=−x+ 4
f(x) =−(x−7)2
reflection in y: reflection in x:
Graphing a translationGraphing a translation
On the same set of axis, graph f(x-1)
The graph shifts one unit rightThe graph shifts one unit right
On the same set of axis, graph f(x-1) Note:
examplesexamples
If f(x)=x2 is shifted to the left 3 places, name the Vertex and y intercept of the resulting graph.
If is shifted down 1 unit, what is the function?
If is shifted right one unit and up 2 units,Name the resulting function
f (x)= x
f (x)= x
examplesexamples
If f(x)=x2 is shifted to the left 3 places, name the Vertex and y intercept of the resulting graph.
f (x)=(x+ 3)2
Vertex: (-3,0)Y-int: (0,9)
examplesexamples
If is shifted down 1 unit, what is the function?
If is shifted right one unit and up 2 units,Name the resulting function
f (x)= x
f (x)= x
f (x)= x −1
f (x)= x−1+2
Stretches and ShrinksStretches and Shrinks
y=2x2
y=12(x+ 3)
y=13(x−7)
y=3 x+ 3
Vertical stretch 2
Vert. shrink of ½
Vertical shrink 1/3
Vert. stretch of 3
Stretches and ShrinksStretches and Shrinks
y=2x2
y=(12
x+ 3)
y=13(x−7)
y=2x+ 3
Vertical stretch 2
Horizontal stretch 2
Vertical shrink 1/3
Horizontal shrink ½
X is still confused!
Circles: center (h,k) radius = rCircles: center (h,k) radius = r
(x−3)2 +(y+2)2 =9
Center: (3,-2) radius = 3
(x−h)2 +(y−k)2 =r2
examplesexamples
1. Center: (-5, 2) radius = 7
Write the equation
2.
What is the center and the radius
x2 +(y+1)2 =64
ReviewReview
Name all transformations Name all transformations
f (x)=−2 x−6 −7
How did the absolute value function change?