Prepared by Doron Shahar
Relations and functions
Prepared by Doron Shahar
Warm-up: page 32
Write the definition of a relation.
A relation is a correspondence between two sets A and B such that
each element of A corresponds to at least one element of B.
The domain of a relation is: The set A.
The range of a relation is: The set B.
Write the definition of a function.
A function is a relation such that each element of the domain
corresponds to exactly one element in the range.
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Relations in diagrams
Domain
Range
Relation
−2
0
1
1
2
3
4
5
Domain
Range
−3
−1
0
1
3
Relation
Is it a function?
Yes
No
Is it a function?
2
Yes
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3.1.1(A) Relations in Tables
Determine whether the following represents y as a function of x.
x
y
Domain
−3
−1
0
1
3
-3
2
-1
2
0
2
1
2
3
2
Range
Is it a function?
2
Yes
If none of the x-values is repeated, then
the table represents y as a function of x.
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3.1.1(B) Relations in Tables
Determine whether the following represents y as a function of x.
x
y
-2
4
-2
5
0
1
1
2
Domain
Range
−2
0
1
1
2
3
4
5
1
3
Is it a function?
No
If at least one of the x-values is repeated, then
the table does not represent y as a function of x.
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3.1.2 Relations in Graphs
A graph represents the graph of a function if and only if no
once
vertical line passes through the graph more than ________.
Determine whether the following represent y as a function of x.
Function
5
4
3
2
1
0
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-2
-3
-4
-5
-6
-7
(A)
Not a function
8
(B)
6
4
2
0
-6
-4
-2
0
-2
-4
-6
2
4
6
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3.1.3 Relations in Equations
Determine whether the following represent y as a function of x.
Steps: 1) Solve for y
2) Plug in trial values for x
• If for some value of x there is more than one y value,
then the equation does not represent y as a function of x.
• If for all values of x there is only one y value,
then the equation represents y as a function of x.
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3.1.3(C) Relations in Equations
Determine whether the following represents y as a function of x.
1) Solve for y
Starting Equation
Solution for y
y   x5
y
x5
2) Plug in trial values for x
If x=5, y   5  5   0   0  0
If x=9,
y   9  5   4  2
There is a value of x, namely x=9, for which there are two
corresponding y values: +2 and −2. y is NOT a function of x.
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3.1.3(A) Relations in Equations
Determine whether the following represents y as a function of x.
1) Solve for y
Starting Equation x 2  y 2  9
2
2
Subtract x2
y 9x
Take square root y 2  9  x 2
Solution for y
y   9 x
2
2) Plug in trial values for x
If x=0, y   9  0   9   3
There is a value of x, namely x=0, for which there are two
corresponding y values: +3 and −3. y is NOT a function of x.
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3.1.3(D): Relations in Equations
Determine whether the following represents y as a function of x.
1) Solve for y
Starting Equation
Divide by x
Take cube root
Solution for y
xy  7
3
y 7 x
3
3
y 
3
y 
3
3
7/x
7 x
2) Plug in trial values for x
If x=−7, y 
3
 1  1
For each value of x, there is only one corresponding y value.
y is a function of x.
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3.1.3(B) Relations in Equations
Determine whether the following represents y as a function of x.
1) Solve for y
Starting Equation xy  4 y  7
y ( x  4)  7
Factor out y
Divide by (x−4)
y 
Solution for y
2) Plug in trial values for x
If x=−3, y 
7
34

7
7
7
x4
 1
For each value of x, there is only one corresponding y value.
y is a function of x.
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3.1.6 Review
(B) Create a table of values for x and y so that y is NOT a function of x.
(C) Create a graph which does NOT represent y as a function of x.
(A) Create an equation in x and y so that y is NOT a function of x.
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Warm-up
Write the following using interval notation.
−13 < x ≤ −5
r ≥ 3.6
y < −2
All real numbers
All real numbers except x = 3
All real numbers except x = −2 and x = 4
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Interval notation
Write the following using interval notation.
−13 < x ≤ −5
Endpoint not included
Endpoint included
  13 ,  5 
r ≥ 3.6
3 . 6 ,   Mnemonic : 3.6 ≤ r < ∞
y < −2
   , 2  Mnemonic: −∞ < y <−2
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Interval notation
Write the following using interval notation.
All real numbers
  ,  
All real numbers except x = 3
   , 3   3 ,  
All real numbers except x = −2 and x = 4
   , 2     2 , 4    4 ,  
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3.2.2 Domain and Range from Graphs
Find the domain and range of the graph shown.
5
4
3
2
1
0
-5
-4
-3
-2
-1-1 0
-2
-3
1
2
3
4
5
Range:
possible y-values
Range :  5 , 4 
-4
-5
-6
Domain: possible x-values
Domain
:  4 , 4 
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Page 33: Polynomial functions
A polynomial function is a function of the form:
a n x    a 2 x  a1 x  a 0
n
2
f(x)=_________________________, where a0, a1, a2, …, an
are real numbers and n is a non-negative integer.
Examples of polynomial functions:
g(x)=3x4−2x3+x2−x
f(x)=3x5+2x3+7x2+9x+12
h(x)=x2+6x+9
q(x)=2x−1
r(x)=17




,

The domain of every polynomial function is___________.
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Page 33: Rational functions
g ( x)
h( x)
A rational function is a function of the form: f(x)=______,
where g and h are polynomial functions and h(x) ≠ 0.
Examples of rational functions:
3x5+2x3+7x2+9x+12
17
s(x)=
t(x)=
4
3
2
3x −2x +x −x
2x−1
The domain of every rational function is
the set of all x except whe re the denominato r equals zero
______________________________________________.
17
Example: The denominator of t(x)=
is zero only
2x−1
when x=1/2. The domain of t(x) is    , 1 2   1 2 ,  
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Page 33: Root functions
n
g ( x)
A root function is a function of the form: f(x)= _________,
where n is an integer such that n ≥ 2.
Examples of root functions:
f(x)= 3 x
f(x)= 4 x
f(x)= x
f(x)= 2−5x
If n is even, the domain of the root function is
the set of all x where g ( x )  0 The domain of f(x)= x is 0 ,  
_________________________.
If n is odd, the domain of the root function is
the set of all x where g ( x ) is defined
_____________________________.
The domain of f(x)= 3 x
is    ,  
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(A) g ( x )  3 x  5 x  4
2
What type of function is g(x)?
A polynomial function
What is the domain of a polynomial function?
What is the domain of g(x)?
(−∞,∞)
(−∞,∞)
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(B) h ( x ) 
2x 1
x3
What type of function is h(x)?
A rational function
Describe the domain of this rational function.
The set of all x except where x+3=0. (x≠−3)
What is the domain of h(x) in interval notation?
   , 3     3 ,  
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(C) T ( x ) 
x2
x  3 x  10
2
What type of function is T(x)?
A rational function
Describe the domain of this rational function?
The set of all x except where x2−3x−10=0. (x≠−2, x≠5)
What is the domain of T(x)?
   , 2     2 , 5   5 ,  
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(D) R ( x ) 
2  5x
What type of function is R(x)?
A root function
Describe the domain of this root function.
The set of all x where 2−5x ≥ 0. (x ≤ 2/5)
What is the domain of R(x) in interval notation?
  , 2 / 5 
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3.1.7 Review
Determine a possible equation for a function with the given domain.
(A) Domain: the set of all real numbers except x=−2
(C) Domain: the set of all real numbers except x=1 and x=−3
(B) Domain: the set of all real numbers greater than or equal to 4
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3.1.4 Evaluating a Function at points
Use the function f(x)=3x2−5x to evaluate the following.
Plug in 4 for x.
(A)
f ( 4 )  3 ( 4 )  5 ( 4 )  3  16  20  48  20  28
2
f ( 3 )  3 ( 3 )  5 ( 3 )  3  9  15
2
 27  15
 12
f ( 0 )  3( 0 )  5 ( 0 )  3  0  0  0  0  0
2
2

3
(

2
)
 5 (  2 )  3  4  10  12  10  22
f ( 2 )
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3.1.4 Evaluating a Function
Use the function f(x)=3x2−5x to evaluate the following.
f (h )  3 ( h )  5 ( h )  3 h  5 h
2
2
f ( 2 h )  3 ( 2 h )  5 ( 2 h )  3  4 h 2  10 h  12 h  10 h
2
(B)
2
f ( h  4 )  3 ( h  4 )  5 ( h  4 )  3  ( h  8 h  16 )  5 h  20
2
2
 3 h  24 h  48  5 h  20  3 h  19 h  28
2

2

f ( h )  4  3h  5 h  4  3 h 2  5 h  4
2
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3.1.4 Evaluating a Function
Use the previous slides to evaluate the following.
f ( 4 )  28
f (h )  3 h  5 h
2

f ( h  4 )  3 h 2  19 h  28

(C)
2
f ( h )  f ( 4 )  3 h  5 h   28   3 h  5 h  28
(D)
f ( h  4 )  f ( 4 )  3 h  19 h  28  28   3 h 2  19 h

2
2

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3.1.4 Evaluating a Function
Use the function f(x)=3x2−5x to evaluate the following.
2
f ( x )  3( x )  5 ( x )  3 x  5 x
2
2
f (  x )  3(  x )  5 (  x )  3  x 2  5 x  3 x  5 x
2
f ( x  2 )  3 ( x  2 )  5 ( x  2 )  3  ( x  4 x  4 )  5 x  10
2
2
 3 x  12 x  12  5 x  10  3 x  17 x  22
2
2
f ( x  h )  3 ( x  h )  5 ( x  h )  3  ( x 2  2 hx  h 2 )  5 x  5 h
2
 3 x  6 hx  3 h  5 x  5 h
2
2