1. Relations and functions
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Prepared by Doron Shahar
Chapter 3 Section 3.1
Relations and functions

2. Warm-up: page 32 Write the definition of a relation.
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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 range of a relation is:
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.
The set A.
The set B.

3. Relations in diagrams Yes No Yes 1 −2 2 3 4 1 5 −3 −1 2 1 3
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Prepared by Doron Shahar
Relations in diagrams
Domain
Range
Relation 1
Is it a function? −2 2
Yes No 3 4 1 5
Domain
Range −3
Relation
Is it a function? −1 2
Yes 1 3

4. 3.1.1(A) Relations in Tables
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3.1.1(A) Relations in Tables
Determine whether the following represents y as a function of x. x -3 -1 1 3 y 2
Domain
Range −3
Is it a function? −1 2
Yes 1 3
If none of the x-values is repeated, then
the table represents y as a function of x.

5. 3.1.1(B) Relations in Tables
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3.1.1(B) Relations in Tables
Determine whether the following represents y as a function of x. x -2 1 y 4 5 2 3
Domain
Range
Is it a function? 1 −2 2 No 3 4 1 5
If at least one of the x-values is repeated, then
the table does not represent y as a function of x.

6. 3.1.2 Relations in Graphs Function Not a function (A) (B)
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3.1.2 Relations in Graphs
A graph represents the graph of a function if and only if no vertical line passes through the graph more than ________.
Determine whether the following represent y as a function of x.
Function
Not a function
(A)
(B)

7. 3.1.3 Relations in Equations
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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.

8. 3.1.3(C) Relations in Equations
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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
2) Plug in trial values for x
If x=5,
If x=9,
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.

9. 3.1.3(A) Relations in Equations
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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
Subtract x2
Take square root
Solution for y
2) Plug in trial values for x
If x=0,
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.

10. 3.1.3(D): Relations in Equations
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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
2) Plug in trial values for x
If x=−7,
For each value of x, there is only one corresponding y value.
y is a function of x.

11. 3.1.3(B) Relations in Equations
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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
Factor out y
Divide by (x−4)
Solution for y
2) Plug in trial values for x
If x=−3,
For each value of x, there is only one corresponding y value.
y is a function of x.

12. Prepared by Doron Shahar
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.

13. Prepared by Doron Shahar
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

14. Interval notation Mnemonic : 3.6 ≤ r < ∞ Mnemonic: −∞ < y <−2
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Interval notation
Write the following using interval notation. −13 < x ≤ −5 r ≥ 3.6 y < −2
Endpoint not included
Endpoint included
Mnemonic : 3.6 ≤ r < ∞
Mnemonic: −∞ < y <−2

15. Prepared by Doron Shahar
Interval notation
Write the following using interval notation. All real numbers All real numbers except x = 3 All real numbers except x = −2 and x = 4

16. 3.2.2 Domain and Range from Graphs
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3.2.2 Domain and Range from Graphs
Find the domain and range of the graph shown.
Range:
possible y-values
Domain: possible x-values

17. Page 33: Polynomial functions
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Page 33: Polynomial functions
A polynomial function is a function of the form: f(x)=_________________________, where a0, a1, a2, …, an are real numbers and n is a non-negative integer.
Examples of polynomial functions:
f(x)=3x5+2x3+7x2+9x+12
g(x)=3x4−2x3+x2−x
h(x)=x2+6x+9
q(x)=2x−1
r(x)=17
The domain of every polynomial function is___________.

18. Page 33: Rational functions
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Page 33: Rational functions
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)=
3x4−2x3+x2−x
2x−1
The domain of every rational function is
______________________________________________. 17
Example:
The denominator of
t(x)=
is zero only
2x−1
when x=1/2.
The domain of t(x) is

19. Prepared by Doron Shahar
Page 33: Root functions
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 domain of
f(x)= x is
If n is odd, the domain of the root function is
_____________________________.
The domain of
f(x)= 3 x is

20. 3.1.5 Finding the Domain of a Function
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(A)
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)?

21. 3.1.5 Finding the Domain of a Function
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(B)
What type of function is h(x)?
A rational function
Describe the domain of this rational function.
(x≠−3)
The set of all x except where x+3=0.
What is the domain of h(x) in interval notation?

22. 3.1.5 Finding the Domain of a Function
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(C)
What type of function is T(x)?
A rational function
Describe the domain of this rational function?
(x≠−2, x≠5)
The set of all x except where x2−3x−10=0.
What is the domain of T(x)?

23. 3.1.5 Finding the Domain of a Function
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3.1.5 Finding the Domain of a Function
Determine the domain of the following function.
(D)
What type of function is R(x)?
A root function
Describe the domain of this root function.
(x ≤ 2/5)
The set of all x where 2−5x ≥ 0.
What is the domain of R(x) in interval notation?

24. Prepared by Doron Shahar
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

25. 3.1.4 Evaluating a Function at points
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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)

26. 3.1.4 Evaluating a Function (B)
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Prepared by Doron Shahar
3.1.4 Evaluating a Function
Use the function f(x)=3x2−5x to evaluate the following.
(B)

27. 3.1.4 Evaluating a Function (C) (D)
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3.1.4 Evaluating a Function
Use the previous slides to evaluate the following.
(C)
(D)

28. Prepared by Doron Shahar
3.1.4 Evaluating a Function
Use the function f(x)=3x2−5x to evaluate the following.