Relations and Functions
A relation is a set of pairs of input and
output values. You can represent a relation in four different ways as shown.
Not allrelation are funcition
Relations and Functions
The domain of a relation is the set of inputs, also called x-coordinates, of the ordered
pairs.
The range is the set of outputs, also called y-coordinates, of the ordered pairs.
s input p
pOutput any y
possible
Relations and Functions
Function -A type of relation where there is exactly one output for every input. For every x there is exactly one y.
Is the relation a function?
I te l many to
function
I to many
N1497 1477
M
functionis notRelations and Functions
A function is a relation in which each element of the domain corresponds with
exactly one element of the range.
You can use the vertical-line test to
determine whether a relation is a function.
The vertical-line test states that if a vertical line passes through more than one point on
the graph of a relation, then the relation is NOT a function.
5
Relations and Functions
If a vertical line passes through a graph at more than one point, there is more than one
value in the range that corresponds to one value in the domain.
6
p
Relations and Functions
Use the vertical line test. Which graph(s) represent functions?
Graph of a function
no amp e g notGraph of a function
Use Vertical line test
to determine if Graph is
Relations and Functions
A function rule is an equation that represents an output value in terms of an input value. You can write a function rule in function notation. Shown
below are examples of functions rules.
The independent variable, x, represents the input of the function. The dependent variable, f(x),
represents the output of the function.
a input of funnier
Function Notation
y 2x 3 f (x) 2x 3
when x 1, y when x 2, y when x 3, y
when x 4, y
f (1) f (2)
f (3) f (4)
M 2 7 3
Replace hwithgiven Value 22 3
Other example:
• Given
• Evaluate:
. 5 ,
8
5 1
, 4 2
5
0 5
, 1 )
(
2
x x
x x
x x
x h
3 2
) 8 (
) 5 ( 8 )
5 (
h
h
h
PracticeMare
Piecewise f
fytica medlarMedical p
r
square
pas ca is
d 97 81317 at E
F
h C S
F
1 2h s Undifine h s 5 5 2 4
31
h 8 868383512
The last thing we need to learn about functions for
this section is something about their domain. Recall domain meant "Set A" which is the set of values you plug in for x.
For the functions we will be dealing with, there are two "illegals":
1. You can't divide by zero (denominator (bottom) of a fraction can't be zero)
2. You can't take the square root (or even root) of a negative number
When you are asked to find the domain of a function, you can use any value for x as long as the value
How to find domain of function
must be realnumber
Find the domain for the following functions:
x 2 x 1
f
Since no matter what value you
choose for x, you won't be dividing by zero or square rooting a negative number, you can use anything you want so we say the answer is:
All real numbers x.
2 3
x x x
g
If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal
because you can't divide by zero.
The answer then is:
All real numbers x such that x ≠ 2.
means does not equal illegal if this is
zero
Note: There is nothing wrong with the top = 0 just means the
fraction = 0
12
gen I has Nti o
n Sato 9
D n 213 Nosolution
Dg In1213
N 2 0
Let's find the domain of another one:
x x 4
h
We have to be careful what x's we use so that the second
"illegal" of square rooting a negative doesn't happen. This means the "stuff" under the square root must be greater
than or equal to zero (maths way of saying "not negative").
Can't be negative so must be ≥ 0
0 4
x
solve thisx 4
So the answer is:
stoopDha IF
p.mg az
a 4
224
É hens
In21
2 420 no
Ming actin
Find the domain of each function:
5 . 1
1
y x
17 4
.
2 y x
9 .
3 y x
99 .
4 y x 2
Nts 20 D
at
s22 5
42
III L
Nta 20
Do a 2 a N 9920
x 2 99
The sum f + g
f g x f x g x
This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.
x 2 x
2 3 g x 4 x
3 1
f
1 4
3
2
2
3
g x x
f
4 2
4
3
2
x x
Combine like terms & put in
descending
The difference f - g
f g x f x g x
To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you
distribute the – to each term of the second function. You should simplify by combining like terms.
x 2 x
2 3 g x 4 x
3 1
f 2
2 3 4
3 1
g x x
f
1 4
3
2
2
3
x x
Distribute negative
2 2
4
3
2
x x
The product f • g
f g x f x g x
To find the product of two functions, put parenthesis around them and multiply each term from the first
function to each term of the second function.
x 2 x
2 3 g x 4 x
3 1
f g 2 x
2 3 4 x
3 1
f
3 12
2
8
5
2
3
x x x
FOIL
Good idea to put in descending order
but not required.
The quotient f / g
x
g
x x f
g
f
To find the quotient of two functions, put the first one over the second.
x 2 x
2 3 g x 4 x
3 1
f
1 4
3 2
3 2
x x g
f
Nothing more you could dohere. (If you can reduce these you should).
Adding and Subtracting Functions
g - f and g
f Find
. 12 2
g and
8 3
f Let
x x x
x
ft g 1 I ate C D SCD 4
fca 3ns
3ex a 8 323ats
i
y Cftg ca fc iltgC D
tcaestgagII 3E.us
32 8 tan 12
Multiplying / Dividing Functions
g Find f
. 1 g
and 1 - f
Let x x2 x x
( )
and f x g x
f
CazIS
f N gcn x I
Domain of (f + g)(x):
Domain of (f - g)(x):
Domain of (f ∙g)(x):
Domain of : {x|g(x)≠0}
g
f D
D
g
f D
D
g
f D
D
) (x g
f
Df Dg
the set of
all real numbers common to the domain of f and g
Domain of (f/g)(x) consists of the
numbers x for which g(x)≠0 that are in the domain of
both f and g
Domain of Function
Let f and g be two functions defined as
) )(
( ) (
) )(
( ) (
x g f b
x g f a
2 ) 1
(
x x
f .
) 1
(
x x x
and g
) ( )
(
) )(
( ) (
g x d f
x g f c
Determine domain of the following.
find domain of f andg
Df at 23 Dg at
for natl goa Fi o
DgMDg NE230 Df NER unto ng
Tgif
Dg x2 13no nCnt2 n at at 2,011
Find
T
Domaingon
ng
on o
You should be familiar with the shapes of these basic
functions. We'll learn them in this section.
Odd and Even Functions
A function f is even if
f(
x) = f(x) for all x in its domain.A function f is odd if
f(
x) =
f(x) for all x in its domain.Vertical line of
reflection at x = 0
Point of rotational symmetry
f N n an
n ens an
not cnaans
2
-7 -6 -5 -4 -3 -2 -1 0 1 3 4 5 6 7 8 7
1 2 3 4 5 6 8
-2 -3 -4 -5 -6 -7
So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.
Even functions have y-axis Symmetry
Constant Functions
f(x) = b, where b is a real number
The domain of these functions is all real numbers.
The range will only be b
f(x) = 3 f(x) = -1 f(x) = 1
Would constant functions be even or odd or neither?
x
f ( x ) b
straightlinepassingthrough
gexist
Equations that can be written f(x) = mx + b
The domain of these functions is all real numbers.
slope y-intercept
Passingthroughyexist
Identity Function
f(x) = x, slope 1, y-intercept = 0
The domain of this function is all real numbers.
The range is also all real numbers
f(x) = x
Would the identity function be even or odd or neither?
If you put any real
number in this function, you get the same real number “back”.
x
f ( x )
linepassingthroughorigin
Square Function
f(x) = x2
The domain of this function is all real numbers.
Would the square function be even or odd or neither?
x
f x n
neven even f
n odd odd f
Cubic Function
f(x) = x3
The domain of this function is all real numbers.
The range is all real numbers
Would the cube function be even or odd or neither?
f ( x )
x
Square Root Function
The domain of this function is
Would the square root function be even or odd or neither?
x x
f
x : x 0
Absolute Value Function
The domain of this function is all real numbers.
The range is
Would the
absolute value function be even or odd or neither?
x x
f
f ( x ) : f ( x ) 0
x
Reciprocal Function
The domain of this function is all NON-ZERO real numbers.
Would the
reciprocal function be even or odd or neither?
x x
f 1
x : x 0
VERTICAL TRANSLATIONS
Basic graph: f
x x2
x y
x y
x 1 x2 1f
x y
x 3 x2 3f
x x
2f
As you can see,a number added or
subtracted from a function will cause a vertical
shift or
translation in the function.
existstay sexistshiftup
byinnit
VERTICAL TRANSLATIONS
Basic graph: f
x x
x 2 x 2f So the graph
f(x) + k, where k is any real
number is the graph of f(x) but vertically shifted by k. If
k is positive it will shift up. If k is negative it will shift down
x y
x y
x y
x 4 x 4
f
x x
f
D NER
R 92watfrom4 a
movingupward
x x2f
x y
As you can see, a number
added or
subtracted from the x will cause
a horizontal shift or
translation in the function but opposite way of
the sign of the number.
HORIZONTAL TRANSLATIONS
x y
x
y
f x x
2
x 1
x 1
2f
x 2
x 2
2f
Basic graph:
HORIZONTAL TRANSLATIONS
x x3f
x y
x y
x
x 1
x 1
3 yf f x x3
x 3 x 33
f
So the graph
f(x-h), where h is any real number is
the graph of f(x) but horizontally
shifted by h.
Notice the negative.
(If you set the stuff in parenthesis = 0 & solve it will tell you how to shift
along x axis).
shift right 3
Basic graph:
x y
x y
x y
We could have a function that is transformed or translated both vertically AND horizontally.
x xf
x x 2f
up 3
left 2
Step 1: Basic graph:
Step 2: Graph:
x x
f
x (x2) f
x (x2) 3
f bysunit
moveleftaunit
mm
D 22 2 RY233
and
If we multiply a function by a non-zero real number it has the affect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.
Let's try some functions from our library of functions multiplied by non-zero real numbers to see this.
DILATION
Basic graph: f
x xSo the graph
a f(x), where a
is any real number GREATER THAN 1, is the
graph of f(x) but vertically
stretched or dilated by a
factor of a.
x y
x y
x y
x x
f
x xf 2
2 4 f x 4 x
Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the
original (blue) graph's y value.
f x
I
kSens k71
basicgraph
So the graph
a f(x), where a
is 0 < a < 1, is the graph of
f(x) but vertically compressed or dilated by a
factor of a.
x y
x y
x y
Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as te original (blue)
graph's y value.
x x
f 4
1 4
1
What if the value of a was positive but less than 1?
x x
f
x x
f 2
1 2
1
Basic graph: f
x xgraphverticallycompress compressp andbecome
furtherthenyexist
x Xx
So the graph - f(x) is a reflection about the x-axis of the graph of f(x).
(The new graph is obtained by
"flipping“ or reflecting the function over the
x-axis)
x y
What if the value of a was negative?
x y
x xf
x x
f
Notice any x on the new (red) graph has a y value that is the negative of the original
(blue) graph's y value.
Basic graph: f
x xNoredreal
Noexpression a realnumber
x y
x x3f
There is one last transformation we want to look at.
x y
Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value.
x x3
f
x x 3f So the graph
f(-x) is a reflection about the y-axis of the graph of f(x).
(The new graph is obtained by
"flipping“ or reflecting the function over the
y-axis)
Basic graph:
Summary of Transformations So Far
horizontal translation of h (opposite sign of number
with the x)
If a > 1, then vertical dilation or stretch by a factor of a
vertical translation of k If 0 < a < 1, then vertical dilation or compression by a factor of a
f(-x) reflection about y-axis
Do reflections BEFORE vertical and horizontal translations
If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)
b k
h f x
a
( )
horizontal dilation by a factor of b
Graph
using transformations
12 1
x
x f
We know what the graph would look like if it was
from our library of functions.
x x
f 1
x y
x y
x y
x y
x y
x x
f 1
x x
f 1 1 1
x x
f 1
2 1
x x
f
t
D n 2
R y 1
The Composition Function
f g x f g x
This is read “f composition g”
and means to copy the f
function down but where ever you see an x, substitute in the g function.
x 2x2 3 g
x 4x3 1f
( )
2 ( ) 3 2
4 1
3) )(
( f g x f g x g x 2 x3 2
5 16
32 3
2 16
32 6 3 6 3
x x x x
1. Evaluate the inner function g(x) first.
2. Then use your answer as the input of the outer function f(x).
g
g f x g f x
This is read “g composition f”
and means to copy the g
function down but where ever you see an x, substitute in the f function.
x 2 x
2 3 g x 4 x 1
f
2 3 1 8 13
4
1 )
( 4
) ( )
)(
(
2
2
x x
x f
x f
g x
f g
1. Evaluate the inner function f(x) first.
2. Then use your answer as the input of the outer function g(x).
f f x f f x
This is read “f composition f” and means to copy the f
function down but where ever you see an x, substitute in the f function. (So sub the function into itself).
x 2 x
2 3 g x 4 x
3 1
f
2 3 3
2
2
2
x
f
f la
Example – Composition of Functions
g f
x g
f
x
x g(x 2) (x 2)2g
49 )
7
(
2
x 2 and
.Find
5f
Let x g x x2 g f
Method 1:
g f
5 5 2
2Method 2:
g f
x g
f
x
f 5
g(5 2)g
49 )
7
(
2
) (7
g
Let’s try some
and
7 .Find
2Let f x x3 g x x2 g f
gof x gtf n
fix 7
x 7
N't 7 gof 2 2 t 7
71
g f a gtfo f 27 238
g 87 8 7
INVERSE
FUNCTIONS
Verifying Inverse Function
Show that each function is an inverse of the other:
3 ) 2
( and 2
3 ) ( . 2
5 and 5 .
1
x x g x
x f
x x g x
x f
Find the inverse of Answer:
x x
f
2
4
2 4
1 2
x
x f
To be a one-to-one function, each y value could only be paired with one x. Let’s look at a couple of graphs.
Look at a y value (for
example y = 3)and see if there is only one x value on the graph for it.
This is a many-to-one function
For any y value, a horizontal line will only intersection the graph once so will only have one x value
This then IS a one-to-one
