Chapter 2: Functions and Graphs PART 1 PART 1
Dec 21, 2015
• A set of ordered pairs (x, y) is also called a relation.
• The domain is the set of x-coordinates of the ordered pairs.
• The range is the set of y-coordinates of the ordered pairs.
Find the domain and range of the relation
{(4,9), (-4,9), (2,3), (10,-5)}
• Domain is the set of all x-values, {4, -4, 2, 10}.
• Range is the set of all y-values, {9, 3, -5}.
Example 1
Example 2
Domain:{Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}
Range:
{20, 15, 10, 7}
• Some relations are also functions.
• A function is a set of order pairs in which each first component in the ordered pairs corresponds to exactly one second component.
Ways to Represent a FunctionWays to Represent a Function
• SymbolicSymbolic
x,y y 2x or
y 2x
X Y
1 2
5 10
-1 -2
3 6
• GraphicalGraphical
• NumericNumeric
• VerbalVerbalThe cost is twice the original amount.
y f x
• Output Value• Member of the Range• Dependent Variable
These are all equivalent names for the y.
• Input Value• Member of the Domain• Independent Variable
These are all equivalent names for the x.
Name of the function
FUNCTION NOTATIONFUNCTION NOTATION
Example 3
y = -3x + 2 so represents a function.
• We often use letters such as f, g, and h to name functions.
• We can use the function notation f(x) (read “f of x”) and write the equation as f(x) = -3x + 2.
Note: The symbol f(x) is a specialized notation that does NOT mean f • x (f times x).
• to evaluate a function at x substitute the x-value into the notation.
• Example 4
f(x) = -3x + 2, f(2) = -3(2) + 2 = -6 + 2 = -4.
Example 5
g(x) = x2 – 2x
a.find g(-3)
b.write down the corresponding ordered pair.
Answer :
• g(-3) = (-3)2 – 2(-3) = 9 – (-6) = 15.
• The ordered pair is (-3, 15).
Drawing Graphs of Functions
A way to visualize a function is by drawing its graphThe graph of a real function f of one variable is the set of all points P(x, y) in the plane such that y = f(x). Plot the value of x on the horizontal, or x-axis and the value of f(x) on the vertical, or y-axis. How can we tell whether a set of points in the plane is the graph of some function? By reading the definition of a function again, we have an answer.
Ex 6.
Given the relation {(4,9), (-4,9), (2,3), (10,-5)}, is it a function?
•Since each element of the domain (x-values) is paired with only one element of the range (y-values) , it is a function.
Note:
Each x-value has to be assigned to ONLY
one y-value!!!
Domain and Range
Is the relation y = x2 – 2x a function?
• Since each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.
Question:What does the graph of this function look like?
Does this graph pass
the vertical line test?
Example 7
8
6
4
2
-2
-4
-6
-10 -5 5 10
f x = x2-x
Is the relation x2 + y2 = 9 a function?
• Since each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function
Check the ordered pairs: (0, 3) (0, -3)
The x-value 0 corresponds to two different y-values, so the relation is NOT a function.
Question: What does the graph of this relation look like?
Example 8
Example 9
Use the vertical line test to determine whether the graph to the right is the graph of a function.
x
y
Since no vertical line will intersect this graph more than once, it is the graph of a function.
Example 10
Use the vertical line test to determine whether the graph to the right is the graph of a function.
x
y
Since no vertical line will intersect this graph more than once, it is the graph of a function.
Example 11
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since a vertical line can be drawn that intersects the graph at every point, it is NOT the graph of a function.
x
y
Example 12
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.
x
y
Find the domain and range of the function graphed (in red) to the right. Use interval notation.
x
y
Domain is [-3, 4]
Domain
Range is [-4, 2]
Range
Determining the domain and range from the graph of a relation:
Example:
Example 13
Find the domain and range of the function graphed to the right. Use interval notation. x
y
Domain is (-, ) DomainRange is [-2, )
Range
Example 14
Find the domain and range of the function graphed to the right. Use interval notation.
x
y
Domain: (-, )
Range: (-, )
Example 15
Find the domain and range of the function graphed to the right. Use interval notation.
x
y
Domain: (-, )Range: [-2.5](The range in this case consists of one single y-value.)
Example 16
Find the domain and range of the relation graphed to the right. Use interval notation.
(Note this relation is NOT a function, but it still has a domain and range.)
Domain: [-4, 4]
Range: [-4.3, 0]
x
y
Example 17
Find the domain and range of the relation graphed to the right. Use interval notation.(Note this relation is NOT a function, but it still has a domain and range.)
Domain: [2]
Range: (-, )
x
y