Unit 1, Part 2 Families of Functions, Function Notation, Domain & Range, Shifting.

Unit 1, Part 2

Families of Functions, Function Notation, Domain & Range, Shifting

Families of Functions

• The following are the names of the basic “parent functions” with which we will be working– Linear– Quadratic– Absolute Value– Square Root– Cubic– Inverse Power

• What is a function?• What is domain?• What is range?• Even function vs Odd function

–Even-fold over y-axis–Odd-fold over origin (y-axis, then x-

axis)

Functions What is a function? What are the different ways to

represent a function?

FunctionA function is a mathematical “rule”

that for each “input” (x-value) there is one and only one “output” (y – value).

A function has a domain (input or x) and a range (output or y)

Examples of a Function

{ (2,3) (4,6) (7,8)(-1,2)(0,4)} 4

-2

1

8

-4

2

4

-2

1

8

-4

2

Non – Examples of a Function

{(1,2) (1,3) (1,4) (2,3)}

Vertical Line Test – if it passes through the graph more than once then it is NOT a function.

You Do: Is it a Function? Give the domain and range of each (whether it’s a function or not).

1.{(2,3) (2,4) (3,5) (4,1)}

2.{(1,2) (-1,3) (5,3) (-2,4)}

3. 4.

5.

0

-3

4

1

-5

9

Function NotationFunction Notation just lets us see

what the “INPUT” value is for a function. It also names the

function for us – most of the time we use f, g, or h.

f(x) = 2x is read “f of x is 2 times x” f(3) = 2 * 3 = 6 The 3 replaced the x for the input.

Function NotationUse the rule: f: “a number times 3

minus 6” to fill in the table for the given inputs:

x f(x)=3x-6 f(x) or y -value

-1

012

Given g: a number squared plus 2

1) Find g(2) 2) Find g(-3)3) Find g(x)4) Find g(2a) 5) Represent g as a mapping for

domain { -2, -1, 0, 1, 2 }

Given f: a number multiplied by 3 minus five

1) Find f(-1) 2) Find f(2)3) Find f(x)4) Find f(3x)5) Find f(x+2) 6) Represent f as a table for domain

{ -4, -2, 0, 2, 4 }

Properties of Functions

• End behavior of a functionAs “x” goes somewhere, where does “f(x) or y” go?

Function Properties…

• Odd degree vs Even degree

Function properties…

• Real 0’s

Shifting Functions

• On your graph paper, graph each parent function.• Graph the following functions (calc, table, however

you’d like).– F(x) = x +3– F(x) = x² + 3 and F(x) = (x + 3)²– F(x) = x³ -2 and F(x) = (x – 2)³ – F(x) = l x l – 4 and F(x) = l x – 4 l– F(x) = √(x) + 1 and F(x) = √(x + 1) – F(x) = 1 and F(x) = 1 - 2

x– 2 x

Shifting continued…

• Looking at the graphs, in small groups see if you can come up with a rule for how graphs are shifted.

Shifting again…

• Use your rule to graph these and describe how they are shifted.– F(x) = x -7– F(x) = (x + 4)² - 2 – F(x) = (x – 2)³ + 6– F(x) = l x – 5 l – 4 – F(x) = √(x + 10) + 3– F(x) = 1 + 3

x– 8

Piecewise Functions

• Give the domain and range of the following function.

Graph the following piecewise functions and give the domain and

range.• f(x) = {x – 4 if x < 2

{ 1 if x > 2• g(x) = { l x + 3l if x < 1

{ x² if x > 2

• f(x) ={ -2 if x > 3 {x + 4 if x < -1

• f(x)={ 2x if x < 4 { lxl+3 if x > -1

Inverses of Functions ( f -1(x) )

• What does “inverse” mean?• Given the following function g(x):

– {(-2,3),(1,7),(3,8),(6,-4)}– Give the domain and range of g(x).– find g -1(x).– Give the domain and range of g -1(x).– Is g -1(x) a function? Why or why not?

Inverses of Functions ( f -1(x) )

• Given f(x) = 4x + 7. How would we find the inverse ( find f -1(x) )?– Step 1: rewrite it as “y = 4x + 7”– Step 2: switch the x and y– Step 3: Solve for y– Step 4: rewrite using function notation