1. 2.1 Functions and Their Graphs

2. What is a Relation?
• A relation is a mapping, or pairing, of input
values with output values.
 “Mapping Diagram”
• The set of input values is the domain.
• The set of output values is the range.
• What are the domain and range of this relation?

3. How Can We Write a Relation?
• A relation can be written in the form of a
table:
• A relation can also be written as a set of
ordered pairs:

4. How Do We Write a Relation with Numbers?
• Set of ordered pairs with form (x, y).
• The x-coordinate is the input and the y-
coordinate is the output.
• Example:
{ (0, 1) , (5, 2) , (-3, 9) }
• { } is the symbol for a “set”
• What is the domain and range of this relation?

5. How Do We Graph a Relation?
• To graph a relation, plot each of its ordered
pairs on a coordinate plane.
• Graph the relation:
{ (0, 1) , (5, 2) , (-3, 9) } Remember:
The x comes first – moves
right or left.
The y comes second –
moves up or down.
Positive means to the
right or up.
Negative means to the
left or down.

6. Your Turn!
• Graph the relation and identify the domain
and range.
{ (-1,2), (2, 5), (1, 3), (8, 2) }

7. What is a Function?
• A function is a special type of relation that
has exactly one output for each input.
• If any input maps to more than one output,
then it is not a function.
• Is this a function? Why or why not?

8. Which of These Relations Are Functions?
•
• { (3,4), (4,5), (6,7), (3,9) }
• X 5 7 9 2 6
y 1 6 2 8 4

9. Using the Vertical Line Test
• A relation is a function if and only if no
vertical line crosses the graph at more
than one point.
• This is not a function because the vertical
line crosses two points.

10. Your Turn!
• Write the domain and range.
• Is this a function?
{ (2,4) (3,6) (4,4) (5, 10) }
stop

11. What is a Solution of an Equation?
• Many functions can be written as an equation,
such as y = 2x – 7.
• A solution of an equation is an ordered pair
(x, y) that makes the equation true.
• Example: Is (2, -3) a solution of y = 2x – 7 ?

12. What are Independent and Dependent
Variables?
• The input is called the independent variable.
▫ Usually the x
• The output is called the dependent variable.
▫ Usually the y
• Helpful Hints:
▫ Input and Independent both start with “in”
▫ The Dependent variable depends on the value of
the input

13. What Does the Graph of an Equation Mean?
• The graph of a two variable equation is the
collection of all of its solutions.
• Each point on the graph is an ordered pair (x, y)
that makes the equation true.
• Example: This is the graph
of the equation y = x + 2

14. How Do We Graph Equations?
• Step 1: Construct a table of values.
• Step 2: Graph enough solutions to notice a
pattern.
• Step 3: Connect the points with a line or curve.

15. Example:
• Graph the equation y = x + 1

16. Your Turn!
• Graph the equation y = x – 2

17. What is Function Notation?
• Function notation is another way to write an
equation.
• We can name the function “f” and replace the
y with f(x).
• f(x) is read “f of x” and means “the value of f
at x.”
▫ Be Careful! It does not mean “f times x”
• Not always named “f”, they sometimes use
other letters like g or h.

18. What is a Linear Function?
• A linear function is any function that can be
written in the form f(x) = mx + b
• Its graph will always be a straight line.
• Are these functions linear?
▫ f(x) = x2 + 3x + 5
▫ g(x) = 2x + 6

19. How Do We Evaluate Functions?
• Plug-in the given value for x and find f(x).
• Example: Evaluate the functions when x = -2.
▫ f(x) = x2 + 3x + 5
▫ g(x) = 2x + 6

20. Your Turn!
• Decide if the function is linear. Then evaluate
the function when x = 3.
g(x) = -3x + 4
Stop?

21. How Do We Find the Domain and Range?
• The domain is all of the input values that make
sense.
▫ Sometimes “all real numbers”
▫ For real-life problems may be limited
• The range is the set of all outputs.

22. Example:
• In Oak Park, houses will be from 1450 to 2100
square feet. The cost C of building is $75 per
square foot and can be modeled by C = 75f,
where f is the number of square feet. Give the
domain and range of C(f).