Chapter 3: Functions and Graphs3.1: FunctionsEssential Question: How are functions different from relations that are not functions?

3.1: Functions•A function consists of:

▫A set of inputs, called the domain▫A rule by which each input determines one

and only one output▫A set of outputs, called the range

•The phrase “one and only one” means that for each input, the rule of a function determines exactly one output▫It’s ok for different inputs to produce the

same output

3.1: Functions

•Ex 2: Determine if the relations in the tables below are functions

a)

b)

Inputs 1 1 2 3 3

Outputs

5 6 7 8 9

Inputs 1 3 5 7 9

Outputs

5 5 7 8 5

Because the input 1 is associated to two different outputs (as is the input 3), this relation is not a function

Because each output determines exactly one output, this is a function. The fact that 1, 3 & 9 all output 5 is allowed.

3.1: Functions

•The value of a function that corresponds to a specific input value, is found by substituting into the function rule and simplifying

•Ex 3: Find the indicated values of a)

b)

c)

2( ) 1f x x 2(3) (3) 1 9 1 10 3.162f

2( 5) ( 5) 1 25 1 26 5.099f

2(0) (0) 1 0 1 1 1f

3.1: Functions

•Functions defined by equations▫Equations using two variables can be used

to define functions. However, not ever equation in two variables represents a function.

▫ If a number is plugged in for x in this equation, only one value of y is produced, so this equation does define a

function. The function rule would be:

3 3

33

3

3

3

352

532

4 2 5 0

4 2 5 0

(4 5) 2

2

2 2

2

2

x y

x y

x y

y

x y

y

x

y

532( ) 2f x x

3.1: Functions

•Functions defined by equations▫ If a number is

plugged in for x in this equation, two separate solutions for y are produced,

so this equation does not define a function.

▫ In short, if y is being taken to an even power (e.g. y2, y4, y6, ...) it is not a function. y being taken to an odd power (y3, y5, y7, …) does define a function

2

2

2

1 0

1 01 1

1

1

1 or 1

y x

y x

y x

y x

y x y

x x

x

3.1: Functions

•Ex 4: Finding a difference quotient▫For and h ≠ 0, find each outputa)

b)

2( ) 2f x x x 2

2 2

( ) ( ) ( ) 2

2 2

f x h x h x h

x xh h x h

2 2

2

22 2

( ) ( ) ( ) ( ) 2 2

22 2

2

f x h f x x h x h x x

x xh h x h

xh h

x

h

x

3.1: Functions•Ex 4 (continued): Finding a difference

quotient▫For and h ≠ 0, find each outputc)

▫If f is a function, the quantityis called the difference quotient of f

2( ) 2f x x x 2( ) ( ) 2

(2 1)

2 1

f x h f x xh h h

h hh x h

hx h

( ) ( )f x h f x

h

3.1: Functions

•Exercises▫Page 148-149▫5-41, odd problems

3.1: Functions

•Domains▫The domain of a function f consists of every

real number unless…1) You’re given a condition telling you

otherwise e.g. x ≠ 2

2) Division by 03) The nth root of a negative number (when n is

even) e.g.

64, , ,...

3.1: Functions

•Finding Domains (Ex 6)▫Find the domain:

When x = 1, the denominator is 0, and the output is undefined. Therefore, the domain of k consists of all real number except 1

Written as x ≠ 1▫Find the domain:

Since negative numbers don’t have square roots, we only get a real number for u + 2 > 0 → u > -2

Written as the interval [-2, ∞)▫Real life situations may alter the domain

2 6( )

1

x xk x

x

( ) 2f u u

3.1: Functions• Ex 8: Piecewise Functions

▫ A piecewise function is a function that is broken up based on conditions

▫ Find f(-5) Because -5 < 4, f(-5) = 2(-5)+3 = -10 + 3 = -7

▫ Find f(8) Because 8 is between 4 & 10, f(8) = (8)2 – 1 = 64 – 1 =

63▫ Find the domain of f

The rule of f covers all numbers < 10, (-∞,10]▫ Discussion: Collatz sequence

2

2 3 if 4( )

1 if 4 10

x xf x

x x

3.1: Functions

•Greatest Integer Function▫The greatest integer function is a piecewise-

defined function with infinitely many pieces.▫ What it means is that

the greatest integer function rounds down to the nearest integer less than or equal to x.

▫ The calculator has a function [int] which can calculate the greatest integer function.

...

3 if 3 2

2 if 2 1

1 if 1 0( )

0 if 0 1

1 if 1 2

2 if 2 3

...

x

x

xf x

x

x

x

3.1: Functions

•Ex 9: Evaluating the Greatest Integer Function▫Let f(x)=[x]. Evaluate the following.

a) f (-4.7) = [-4.7] =b) f (-3) = [-3] =c) f (0) = [0] =d) f (5/4) = [1.25] =e) f (π) = [π] =

-5-3

01

3

3.1: Functions

•Exercises▫Page 148-149▫43-71, odd problems