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 vertical line test-any vertical line drawn will pass through the function in at most one point.

Function notation f(x) = x + 8 Find f(2) This means “let x = 2” f(2) = = 10

Function notation f(x) = 2x - 18 Find f(-1) Find f(2a) Find f(b+3)

Function notation f(x) = 2x - 18 Find f(-1)= -20 Find f(2a) = 2(2a)-18=4a-18 Find f(b+3)=2(b+3)-18=2b+6-18=2b-12

Find: f(-1) and f(3) f(x)

f(-1)=2 and f(3)=2 f(-1)=2 and f(3)=2

Ordered pair examples S={(-3,5),(4,7),(0,5)} S is a function P={(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 function

Which is not a function?

equations In general, x=c are not functions, and relations that have a “y 2 ” are not functions Which are functions?

answers All but these two are functions This is a circle This is a vertical line

One 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 One to one and onto Not one to one or onto D RDRDR

Graphs of one to one Which is not a function? Which are not one to one?

Equations are they one to one? 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: 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

Is this a one to one function? domainRange Make a list of ordered pairs….

List all x’s and list all y’s domainRange

examples S = {(-3,5),(4,7),(0,5)} Domain = {-3,4,0} Range = {5,7}

Restricted domains-denominators and radicals! Set denominator =0 and solve Set and solve Set x + 1>0 and solve

Answers to domains X - 9=0 X=9 D: x + 1>0 D: x >-1 D:

Range Look at the graph from the bottom up And state the y values: Y=x 2 +2 The lowest point Is (0,2)

Range Look at the graph from the bottom up And state the y values: Y=x 2 +2 Range:

assessment State the domain and range of the following Functions:

Domain for both is all reals! State the domain and range of the following Functions:

In interval notation Find the domain and range:

answers Domain: [-5,4] Range: [-4,3]

answers D: R:

answers D: R:

compositions A function inside a function-above: the g function Is placed inside the f function. Find g(3) Take that answer and find F(that answer)

compositions A function inside a function-above: the g function Is placed inside the f function. g(3)=9 Take that answer and find f(9)=9+1=10

compositions A function inside a function-above: the g function Is placed inside the f function. Steps: 1)write f: x+ 1 2)Replace x: ( ) + 1 3)Put g in ( ): (x 2 ) + 1 4)Simplify: x 2 + 1

examples

Examples-answers

Do and hand in Given:

Inverses f -1 (x) If a function is one to one then it is an inverse function. “Swap” x’s and y’s. The graph of a function 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. Given f(x) = 2x + 4, find the inverse… “swap and solve”: Y= SWAP X & Y SOLVE FOR Y Simplify Change y to f -1

examples Find the inverse:

examples Y=x Y=1/3x+3 Y=3x-9

Example 3 Swap x & y Cross multiply Get y’s together Factor out y Solve for y

f(f -1 (x))=x We call the identity function y=x When we compose inverses of each other, in either order, we get x as an answer. Example: Distribute the 3

Restricting domains When a function is not one to one, we can find a solution if we restrict the domain So now we are only considering half of our graph. and can find the inverse.

solution Y=x

practice Given:

Transformations of functions Vertical shifts: F(x) + a is a shift up a units F(x) – a is a shift down a units Horizontal shifts: x+ a is a shift to the left a units x – a is a shift to the right a units So x is confused and y is not!

Library of functions Y = x 2

examples Describe the shifts of the functions:

examples Horizontal right 8 Horizontal left 1 Vertical down 6 Horizontal left 5 and vertical up 2

reflections A reflection in the y axis is a negation of x A reflection in the x axis is a negation of y (or f(x)) Examples: reflection in y: reflection in x:

Graphing a translation On the same set of axis, graph f(x-1)

The graph shifts one unit right On the same set of axis, graph f(x-1) Note:

examples If f(x)=x 2 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

examples If f(x)=x 2 is shifted to the left 3 places, name the Vertex and y intercept of the resulting graph. Vertex: (-3,0) Y-int: (0,9)

examples If is shifted down 1 unit, what is the function? If is shifted right one unit and up 2 units, Name the resulting function

Stretches and Shrinks Vertical stretch 2 Vert. shrink of ½ Vertical shrink 1/3 Vert. stretch of 3

Stretches and Shrinks Vertical stretch 2 Horizontal stretch 2 Vertical shrink 1/3 Horizontal shrink ½ X is still confused!

Circles: center (h,k) radius = r Center: (3,-2) radius = 3

examples 1. Center: (-5, 2) radius = 7 Write the equation 2. What is the center and the radius

Review

Name all transformations How did the absolute value function change?