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?

Function A 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)}

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)}

Function Notation Function 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 Notation Use the rule: f: “a number times 3 minus 6” to fill in the table for the given inputs: xf(x)=3x-6 f(x) or y -value 0 1 2

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 function As “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) = 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) = 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