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Chapter 3: Functions and Graphs 3.1: Functions. Essential 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 - PowerPoint PPT Presentation
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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
•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