1. 3.1 Functions and their Graphs
Relation – a mapping, or pairing of input values with output values.
Domain – set of input values
Range – set of output values

2. Functions
Function – a relation is a function if there is exactly one output for each input.

3. Relations and Functions
Relations and functions between 2 quantities can be represented in many ways:
-mapping diagrams
-tables
-graphs
-equations
-verbal descriptions

4. Functions and Relations
Relations can be represented by ordered pairs (x, y) where x-coordinate is the 1st number and y-coordinate is the 2nd number.
Domain = First number (input)
Range = Second number (output)

5. Functions and Relations
y-axis
Quadrant I
Quadrant II
x-axis
Quadrant III
Quadrant IV

6. Relations and Functions
Consider the following points:
{(a, 1), (b, 2), (c, 3), (e, 2)}
List the domain: {a, b, c, e}
List the range: {1, 2, 3}

7. Relations and Functions
Ex 2: {(3, 5), (4, -6), (2, -4), (-1, 5)}
List the Domain:
{-1, 2, 3, 4}
List the Range:
{-6, -4, 5}

8. Relations and Functions
How to tell if a relation is a function:
-Only one output for each input (no x can be repeated)
-Vertical Line Test: = no vertical line intersects the graph of the relation at more than 1 point.

9. Relations and Function
For the relation to be a function, no x may be repeated
Are the following Functions?
1. {(1, 3), (-4, 2), (-6, 2), (0, 5)}
Yes = no x has been repeated
2. {(1, 3), (-4, 2), (-6, 7), (1, 5)}
No = 1 was repeated

10. Relations and Functions
Input Output
Age Weight
Write as Ordered Pairs
(16, 220), (16, 122), (17, 179)
(18, 125), (18, 116)
116 Not a function!!

11. Relations and Functions
Input Output
Name Weight
Sue
Mary 133
Steve 159
Carol 144
Jose
Write as ordered pairs:
{(Sue, 125), (Mary, 133), (Steve, 159), (Carol, 144), (Jose, 133)}
Yes, it is a function – no Input has been repeated

12. Relations and Functions
Vertical Line Test
Is this a Function?
Yes

13. Vertical Line Test, cont
Are the following functions? No

14. Relations and Functions
Many functions can be represented by an equation in 2 variables:
Ex: y = 2x – 7
An ordered pair (x, y) of the equation is a solution of the equation if the ordered pair is true when the values of x and y are substituted into it.

15. Relations and Functions
Ex: for the line y = 2x – 7, is the ordered pair (2, -3) a solution?
Substitute the values in for x and y
-3 = 2 (2) – 7
-3 = YES, the ordered pair is a solution of the equation.

16. Relations and Functions
Are the following solutions to the equations?
y = 3x – 1 ; ((2, 5), (-2, -7)
yes, yes
2. 2p + q = 5; (2, 3) (-5, 15)
no, yes

17. Relations and Functions
In an equation, the input variable (x, domain) is the independent variable, and the output (y, range) is the dependent variable because it depends on the value of the input.

18. 3.3 Functions - Continued
Function Notation – the symbol f(x) is read “f of x” and is used to notation a function.
Since a function is a relation, a function can be listed as a set of ordered pairs
(x, f(x))
where the domain is all values for which the function is defined, and the range consists of the values of f(x) where x is the domain of f.

19. Functions, Cont
To determine the range of a given function (given the domain), simply plug the values in for the variable.
Ex: f(x) = 3x + 2 Domain: {-1, 0, 5}
f(-1) = 3(-1) + 2 = -1
f(0) = 3(0) + 2 = 2
f(5) = 3(5) + 2 = 17

20. Functions, Cont Find the range of f(x) = 2x – 7
given the D {-3, -1, 0, 7}

21. Functions, cont Find the Domain of x:
We assume the domain of a function to be all real numbers that are an acceptable replacement for the variable (x).
To find the domain of a function, we must determine whether there are any unacceptable replacements.

22. Unacceptable Replacements
2 Things that make unacceptable replacements:
0 in the denominator – if a value would make the denominator = 0, then the value is unacceptable.
(-) under the radical – if a value would cause the expression under the radical to be a negative number, then the value would be unacceptable.

23. Domain of a FUNCTION Therefore the domain of the function is
Find the domain of:
f(x) =
What happens if x = -3?
f(-3) = = Undefined
Therefore the domain of the function is
D = {x| x 3}
Which reads all x such that x does not equal -3

24. Domain of a function
Find the domain of the following functions:

25. Homework
p. 109 (13-20)
p. 114 (9, 11, 19)
p. 119 (1-27 odd)