Lecture 3.1 to 3.2 bt

Today’s Agenda

Attendance / Announcements

◦ Projects due at Start of Class

◦ Quiz Wednesday

◦ Need Graphing Calculators (TI-83,84,..)

Section 3.1, 3.2

Functions

A function is a rule (think: operation) that assigns an input value to exactly one specific output

f(x) can be thought of as “y”

Examples…before we really define what a function is.

32 xy 32)( xxf

xxy 323 2 ttth 323)( 2

xy 3 xxg 3)(

A function is a rule (think: operation) that assigns an input value to exactly one specific output

inputs outputsf(x)

Domain Range

Determine if each relation is a function or not…

𝐢𝐧𝐩𝐮𝐭 𝒙 9 4 1 0 1 4 9

𝐨𝐮𝐭𝐩𝐮𝐭 𝒚 3 2 1 0 - 1 - 2 - 3

“Functions need to be predictable!”

Determine if each relation is a function or not…

𝐢𝐧𝐩𝐮𝐭 𝒙 -2 -1 0 3 4 6 9

𝐨𝐮𝐭𝐩𝐮𝐭 𝒚 3 2 1 0 1 2 3

Since it is a function, it has a domain and range:

Domain (set of all inputs): { }

Range (set of all outputs): { }

Function Notation (p. 137)

Evaluating Functions

,2)( 2 xxxfif

)3(f

)3(f

)(tf

)3(xf

“The Difference Quotient”

The difference quotient for a function f(x) is:

h

xfhxf )()(

Substitute, Subtract, and Simplify

h

xfhxf )()(

Find the difference quotient of:

xxf 2)(

h

xfhxf )()(

Find the difference quotient of:

53)( xxf

Finding Domains of Functions

The domain of a function is the set of values where the function is defined.

So, to find domains, we need to think about where functions are NOT defined!

Finding Domains of Functions

Red Flags

• Zero(s) in the denominator• Negative under square root

Finding Domains of Functions

𝑓 𝑥 = 2𝑥 + 5 2

Finding Domains of Functions

𝑝 𝑥 = 5 − 𝑥

Finding Domains of Functions

𝑔 𝑥 =𝑥

𝑥2+𝑥−2

Piecewise Functions

01

01)(

2

xx

xxxf

“Pieces of the

Function”

Domain of each piece

A single function, f

Evaluating Piecewise

01

01)(

2

xx

xxxf

)2(f )0(f )4(f

Piecewise Functions

14

132)(

xx

xxxf

Absolute Value Function

xxf )(

Can also be expressed as:

0

0)(

xx

xxxf

Absolute Value Function

0

0)(

xx

xxxf

Domain?

Graphing FunctionsMethod 1: Plotting Points (T-Chart)

3)( xxg

Find the domain of

the function, before

making the T-Chart

Graphing FunctionsMethod 1: Plotting Points (T-Chart)

3)( xxg

Graphing FunctionsMethod 1: Plotting Points (T-Chart)

xxf 2)(

Reading Graphs of Functions(Similar to what we’ve already done)

1 2 3 4

1

2

3

4

)(xf

)2(f

)1(f

)2(f

)4(f

Reading Graphs of Functions(Similar to what we’ve already done)

1 2 3 4

1

2

3

4

)(xf

xxf ,3)(

xxf ,0)(

Positive / Negative graphsIndicate using Intervals

1 2 3 4

1

2

3

4

)(xf0)( xf

0)( xf

Increasing / Decreasing FunctionsIndicate using Intervals

1 2 3 4

1

2

3

4

)(xf.)( incxf

.)( decxf

Graphing Functions

Determining if a graph is a function

The Vertical Line Test

This is a visual representation of what we did last class…Each input cannot have more than one output (not predictable)

Determining if a graph is a function

The Vertical Line Test

Classwork

• Go over worksheets as class

Classwork / Homework

Page 140

9 – 19 odd,

23 – 29 odd,

35 – 45 odd, 51

Graphing Worksheet