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Functions
Algebra of Functions
Functions

What are functions?
Functions
What are functions?
 Possible answers

Functions are relations in which no
two ordered pairs have the same xcoordinate
 A set of rules for generating a specific
output given an input

Linear Algebra, Abstract
Algebra, Operator Algebra,
Group Theory…
Functions, or subsets of functions, may be
treated as entities on which we perform
operations, much like numbers
 Suggested Reading: Mario Livio, The
Equation That Couldn’t Be Solved


Deals not with functions, but with
equations. This is related, since most
functions we discussed are defined by
equations.
Today

Operations with Functions
Addition and Subtraction
 Multiplication and Division
 Composition

Addition and Subtraction

The domain of the
sum or difference
of functions is the
intersection of the
two domains
( f  g )( x)  f ( x)  g ( x)
That is, the
function must
exist for both f
and g
( f  g )( x)  f ( x)  g ( x)

d f g  d f  dg
d f g  d f  dg
Multiplication and Division
( f g )( x)  ( fg )( x)  f ( x) g ( x)
d fg  d f  d g

For division only,
the quotient
function does not
exist for any values
of x for which the
denominator is 0
f 
f ( x)
  ( x) 
g ( x)
g
d f   d f  d g \{x : g ( x)  0}
 
g
Note on Domain Restrictions
If the domain must be restricted in
order for the function to exist, then
that restriction must be noted! It is a
trait of the function.
 Sometimes, a domain restriction is
implied, but you should steer clear of
that ambiguity

Composition of Functions

Apply the internal function
(f), then apply the external
function (g) to that result
( g f )( x)  g[ f ( x)]
if : rf  d g

The range of f must be
contained in the domain of g
If this is the case, then the
domain of the composition is
the domain of f
then : d g

f
 df
Practice

Given the functions f and g, evaluate each
of the following:
f ( x)  x  1
g ( x)  x 2
( f  g )(4), ( fg )( ), ( f g )( 5), ( g f )(10)
x : ( g
x : ( f
f )( x)  0
g )( x)  19
Does _( f g )( x)  ( g f )( x) ?
Activity, Exercise 5.4.1
Thursday

4, 5, 7, 8, 9
Functions
Compositions
Domain and Range considerations
Identity and Inverse
Today
Compositions
 Identity and Inverse

Mapping Diagrams for
Compositions (P.O.D.)
All problems refer to the general function “gof”
 Draw Mapping Diagrams for compositions in which
The range for f is the same as the domain for g
 The range of f is a subset of the domain of g
 The domain for g is a subset of the range of f
 The range for f and the domain for g are disjoint


Challenge: Can you come up with examples for f
and g that correspond to each diagram?
The Point of that Exercise

Mapping Diagrams


An important tool in all algebras and other fields
Domain/Range requirements (two ways to say it)
The image under the first mapping must belong to
the domain of the second mapping
 The range of the first function must be contained
in the domain of the second function

Does a Composition Exist?
Tools for testing

Domain/Range tables
The is one way to display the range
and domain of each function
 Remember the goal: in order for a
composition to exist, the range of the
first function must be contained in
the domain of the second function

Example



Determine the implied
domains and ranges,
and create a
domain/range table
Does the composition
exist?
If so, determine the
equation
f ( x)  x  9
g ( x)  ln x
Find : ( f g )( x)
Find : ( g f )( x)
2
domain
f
g
range
Note on Existence
You should have determined that the
second composition does not exist
because a logarithm of anything less
than or equal to zero is undefined,
BUT…
 The domain of the quadratic function
can be restricted so that its range is
all positive

Composition Review
Do you know what a mapping diagram is?
 Do you know what a composition is?
 Do you know how to evaluate compositions?
 What has to be true about domains and
ranges in order for a composition to exist?
 Do you know how to restrict a domain to
allow a composition to exist?

Problem to show me that
you know these things
Page 154, 3x
 If one of the compositions does not
exist, show how you could restrict the
domain of the first function to allow it
to exist

Activity, Exercise 5.4.1
more problems
1, 2
 3 (parts 5, 10, 13, 16)


If the composition does not exist, can
you restrict a function’s domain in
order to make it exist?

5, 8, 12, 14, 16, 17, 21

Done last class?

4, 5, 8, 9, 14
Functions
Inverse Functions
Today
Problem of the Day
 Some homework review
 Review of Compositions
 Identity and Inverse Functions

Domain and Range limitations
 Graphing Inverse functions by hand
 Graphing Inverse functions by GDC

Problem of the Day




Show that the composition
does not exist
Restrict the domain of g to
allow it to exist
Give the equation for the
composition
Determine the
composition’s inverse
f ( x)  2 ln( x  5)
g ( x)  x  2
2
( f g )( x)  ?
Identities and Inverses:
Review

Addition



What is the Identity for
Addition?
For a given number a, what
is the additive inverse?
Multiplication


What is the Identity for
Multiplication?
For a given number a, what
is the multiplicative
inverse?
Addition
aI a
1
a  (a )  I
Multiplication
aI  a
1
a(a )  I
Identity and Inverse

Identity Function
I ( x)  x
or
yx

Inverse Function


Implies the “reverse”
operations
Name: “f inverse”
1
f [ f ( x)]  x
Domain and Range Limits
If the inverse function of f (“f inverse” or
“f -1”) exists, then:
 f must be a one-to-one function
 The domain of f is equal to the range
of f -1
 The range of f is equal to the domain
of f -1
Obtaining the Equation for
Inverse Functions
y  e  2x 1



Switch the
independent and
dependent
variables
Isolate the NEW
dependent variable
Note that ordered
pairs are “reversed”
 x  e  2 y 1
 x  e  2 y 1
 ( x  e)  2 y  1
2
 ( x  e)  1  2 y
2
( x  e)  1

y
2
2
Using a Calculator
To graph an inverse function, use the
Draw menu:
 DrawInv [type the equation]

Graphing by Hand
Graph the function f
 Graph the Identity function I(x) = x
 Treat I(x) as an axis of symmetry for
the reflection of the function f


Note: The point (a,b) becomes (b,a)
Graphing Practice


Graph each
function on the
right
Graph the inverse
function by hand


Check with your
calculator
Determine the
inverse function
algebraically
d ( x)  2 x  6
f ( x)  2  1
x
h( x )  x  1
j ( x)  ln x
Functions vs. Relations
Note that the functions should be oneto-one in order to have an inverse
function
 Graph the inverse of y = x2


Is it a function?
Exercise 5.4.2

Don’t do these (save for next class)  3, 5
(a-d)
Homework below:
 6, 8 (left column on each)
 10, 18
Draw a mapping diagram for a composition “g
of f” such that the range of f and the domain
of g intersect
5.4.2
5 (a, e, h)
 18

Functions
Transformations
Functions, So Far
Graphing Exponential and Logarithmic
Functions
 Operations with Functions
 Identity and Inverse Functions
 Compositions


This week: Transformations
Activity


Use your GDC to
graph the functions
on the right
Describe (in words)
how the functions
on the right differ
from the function
y=x2
y  x 4
2
y  ( x  4)
2
y  x  6x  9
2
y  2x
2
y  2x  3
2
y  2( x  3)  1
2
y  a ( x  h)  k
2
Transformations
Transformations of graphs are
changes in location, shape or
orientation
 The types of transformations are:

Translations (horizontal and vertical)
 Dilations (horizontal and vertical)
 Reflections (about the x- and y-axes)

Horizontal Translations

A horizontal translation moves the graph “a”
units to the right
f ( x)  f ( x  a)
Example :
f ( x)  x  f ( x  2)  ( x  2)
2
2
Vertical Translations

A vertical translation moves the graph “b”
units up
f ( x)  f ( x)  b
Example :
f ( x)  x  f ( x)  3  x  3
2
2
Combined Translations


A horizontal translation moves the graph “a”
units to the right
A vertical translation moves the graph “b”
units up
f ( x)  f ( x  a)  b
Example :
f ( x)  x  f ( x)  ( x  2)  3
2
2
Vector Translations
(Notation)

Each point has
coordinates (x,y), which
can be written as a
column vector (x on top, y
on bottom)

Vector translations can be
described the same way
x
 
 y
a
 
0
0
 
b
Vector Translations (cont.)

“x-prime” and “y-prime” are the new x and y
coordinates after the transformation (which,
in this case, is a translation)
 x'  x  a
Horizontal :        
 y '  y   0 
 x '   x  0
Vertical :        
 y '  y   b 
 x'  x  a
Combined :        
 y '  y   b 
Labeling Graphs

The MINIMUM for labeled graphs
Title (the function)
 Label your axes
 Label axes intercepts
 Label the asymptote, vertex or any other
important trait of the graph
 Use at least one other point to indicate the
shape of your graph

• The more points you use, the easier it will be to fit
a curve
Practice with Translations

Exercise 6.1
1 (a and c)
 3c
 4c
 10d

Translations Homework

Translations (6.1)
6, left column
 8 (left column)
 9 (a, b, d only)
 10 (c only)


Graph paper is required
Dilations
Dilations can be in the form of a
Stretching or Shrinking of the graph
 They may also be “in the y direction”
or “in the x direction”



This corresponds to “vertical” or
“horizontal”
Note on the textbook’s language
(“from” axes)
Dilations in the y direction

A dilation in the y direction multiplies all ycoordinates by p


If IpI is greater than 1, this is stretching
If IpI is less than 1, this is shrinking
f ( x)  p f ( x)
Example :
f ( x)  x  2 f ( x)  2 x
2
2
Dilations in the x direction

A dilation in the x direction multiplies all xcoordinates by (1/q). Note that “q” is the reciprocal
of the coefficient


If the coefficient is greater than 1, this is shrinking
If the coefficient is less than 1, this is stretching
f ( x)  f  qx 
Example :
x  x
f ( x)  x  f ( )   
3 3
2
2
How to graph dilations
Graph the original function
 For dilations along the y-axis



Multiply each y-coordinate by p
For dilations along the x-axis
Multiply each x-coordinate by (1/q)
 Note that is x has a coefficient that is
greater than 1, “q” is the reciprocal of
that coefficient

Combined Transformations
In general, follow the order of operations
(PEMDAS), reversing operations and the
order when IN the function
 That means that the transformations fall in
this order (from “deal with them first”)

Dilations along the x-axis
 Horizontal translations (this may go first)
 Dilations along the y-axis
 Vertical translations

Dilation/Translation
Combinations

Given the combined
transformation, the form on
the very bottom is preferred
Challenge: Complete the
sentence below:
 A horizontal dilation by a
factor of 1/5 is equivalent to
a vertical dilation by a factor
of ___
yx
2
y '  (5 x  10)
2
y '  5( x  2)
2
Practice with Dilations

Exercise 6.2
2b
3
6


Graph paper is required
Today
Practice/Review of Translations and
Dilations
 Homework Questions
 Reflections
 Odd and Even Functions

Reflections
Reflections involve transferring the
points on a graph across a line, or
“reflecting a graph about a line”
 Today we will discuss reflections
across the x- and y-axes


Note: how does one draw the graph of
an inverse function
Reflections about the x-axis

A reflection about the x-axis involves
changing the sign of all y-coordinates
f ( x)   f ( x)
Example :
f ( x)  2  f ( x)  2 , not (2) !!!
x
x
x
Reflections about the y-axis

A reflection about the y-axis involves
changing the sign of all x-coordinates
f ( x)  f (  x)
Example :
f ( x)  2  f ( x)  2
x
x
Combined Transformations
In general, follow the order of operations
(PEMDAS), but reverse the order within the
parentheses
 That means that the transformations fall in
this order (from “deal with them first”)

Horizontal translations
 Dilations along the x-axis
 Dilations along the y-axis
 Vertical translations


Note: Reflections take effect with
coefficients
Practice with Reflections

Exercise 6.3


1 (a and c)
3 (a, L*, o)
• Start with the parent function and build the graph
through transformations. Check with your
calculator.

Challenge:

In general, what graph would you obtain if
you were to reflect y = f (x) about the line y =
-x ?
This Week

Next Class
Complete the Transformation unit
 Start to review functions


Thursday, there will be an
assessment:
Operations with functions
 Composite functions
 Inverse functions
 Transformations
 (Note: this is 5.4 through 6.3)

Test

Functions

Familiarity with different types,
domain, range
Operations with Functions
 Composite Functions
 Inverse Functions



Defining and graphing them
Transformations

Translations, Dilations, Reflections
Geometers’ Sketchpad
Activity



Plot a parent function
Choose values for a, b, c, and d (both
positive and negative)
Plot the transformations, one at a time, in
the correct order, to get to the final function
a f (bx  c)  d
Review Exercises

Operations and Composites


Inverse Functions


5.4.1: 3*, 6, 7, 14
5.4.2: 1*, 3, 5*, 6*, 12
Transformations

6.3: 3 (left column), 4 (left column)
Revision Set A (page 233)

2, 4, 6, 7, 9a, 14, 15

On #7, could you combine the
transformations from a, b and c? How
about a, b, c, and d?