Unit 4: Introduction to Functions Name_____________________________

Objectives of Unit 4

Determine whether a relation is a function. Definition of a function Notation for functions 1:1 functions

Represent a function multiple ways Graph ↔ Equation ↔ Table

Graph and transform functions. Translations Reflections Stretches & Compression

Analyze a function to determine its properties.

Domain & Range Evaluations & Operations Visual Properties Algebraic Properties Compose Functions Inverse Functions

FUNCTION:

Example: Function or Not?

FUNCTION NOTATION:

Example: If you have an equation and know it’s a function, you rewrite it like this:y=2x+1 becomes: f=2b becomes:

G=R2+2 becomes: m=√t becomes:Example: If y ( x )=3 x2−4, find:

y (2) y (−3) y (10)

ONE-TO-ONE (1:1):

Example: 1:1 or not?

PARENT FUNCTIONS (COMMON FUNCTIONS):

Constant Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Linear Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Quadratic Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Square Root Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Cubic Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Cube Root Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Absolute Value Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

Rational Function: Table:

x y1:1? Even, odd, or neither?

Domain: Range:

Minimum: Maximum:

Highlight increasing/decreasing/constant intervals X-intercept: Y-intercept:

TRANSFORMATIONS:

Moving Up and DownUp: Notation:

Example: To move the function y=|x| up 3 units, the equation turns into:

Down: Notation:

Example: To move the function y=|x| down 3 units, the equation turns into:

Moving Left and RightLeft: Notation:

Example: To move the function y=|x| left 3 units, the equation turns into:

Right: Notation:

Example: To move the function y=|x| right 3 units, the equation turns into:

ReflectionsVertical (over x-axis): Notation:

Example: To reflect the function y=|x| over the x-axis, the equation turns into:

Horizontal (over y-axis): Notation:

Example: To reflect the function y=|x| over the y-axis, the equation turns into:

Stretches and CompressionsStretch: Notation:

Example: To stretch y=|x| by 3, the equation turns into:

Compress: Notation:

Example: To compress the function y=|x| by 1/3, the equation turns into:

DOMAIN:

Example: Find the domains of the following functions.y=x+4

y=|x−1| y=−√x

RANGE:

Example: Find the ranges of the following functions.

y=x+4

y=|x−1| y=−√x

LOCAL MAXIMUMS:

LOCAL MINIMUMS:

INCREASING INTERVALS:

DECREASING INTERVALS:

CONSTANT INTERVALS:

Example: Using 5 different colors, label all local maxima, local minima, increasing intervals, decreasing intervals, and constant intervals on this graph.

Example: Use a graphing calculator to find the local maximum and minimum of the equation y=x3+3 x2−6 x−5

EVEN FUNCTIONS:

ODD FUNCTIONS:

“NEITHER” FUNCTIONS:

Example: Identify the following functions as even, odd, or neither.y=x2+3

y=x3 y=|x+1|

X-INTERCEPTS:

Y-INTERCEPT:

Example: Find all intercepts of the following functions.

y=x+5 y=√x−4

ANALYZING THE WHOLE FUNCTION:

Function (Include Multiple Representations) Properties/TasksEXAMPLE 1

f ( x )=|x+2|−1Domain:

Range:

1:1?

Intercepts:

Odd/Even/Neither:

Increasing intervals:

Local minimum:

f (4 )=¿

f (20 )=¿

EXAMPLE 2x g(x )

−3 −13

−2 −12

−1 −1

−12

−2

0 ∅

12

2

1 1

2 12

3 13

Domain:

Range:

1:1?

Intercepts:

Odd/Even/Neither:

Local maximum:

Decreasing intervals:

g (8 )=¿

g (0 )=¿

EXAMPLE 3h ( x )=¿

Domain:

Range:

1:1?

Local maximum:

Decreasing intervals:

Intercepts:

Odd/Even/Neither:

h (−4 )=¿

h (4 )=¿

EXAMPLE 4The sophomore class is selling cookie dough to raise funds. For each tub of dough sold, $2 is earned in profit; however, the class must pay an initial cost of $100 to the vendor for handling fees. Find a function for profit, P(x ) in terms of the number of tubs, x sold.

Domain:

Range:

1:1?

Intercepts:

Local maximum:

Increasing intervals:

P (20 )=¿

Odd/Even/Neither:

COMPOSITION OF FUNCTIONS:

Example: Complete the following compositions of functions using the four functions below.

f ( x )=|x+2|−1 g ( x )=1x

h ( x )=−x2+3 p ( x )=2x−100

h (2 )+ p(3) g (4 )− f (−5)

g (3)f (4)

f (−5 ) ∙ h (−1 )=¿

h ( x )−p ( x )=¿ g ( x ) ∙ h ( x )=¿

h ( p (2 ) )=¿ g (f (5 ))=¿

g ( p ( x ) )=¿ ( f ∘ g ) (1 )=¿

INVERSE FUNCTIONS:

How to find them algebraically:

Examples: Find the inverses of the following functions.y=3 x−4 y=x3+5 y=√x−1

How to find them on a table:

Example: Find the inverse of the following table.

How to find them on a graph:

Example: Find the inverse of the following graph.

PIECEWISE FUNCTIONS:

x f(x)-3 -27-2 -8-1 -10 01 1

2 83 27

Example: Analyze the following piecewise function.b ( x )={ x+1if x<0( x−1 )2if x≥0} A) f (−1 )=¿

B) f (3 )=¿

C) Domain:

D) Range:

E) Highlight any increasing intervals

F) Is it 1:1?

G) Even, odd, or neither?

Example: Graph the following piecewise functions.f ( x )={x if x<04 if x≥0} g ( x )={ |x+1|if x≤1

−( x−1 )2+2if x>1} h ( x )={x−3 if x←25if x=−2−x if x>−2}