Transcript Function

Section 1.2
Functions and Graphs
Relation
A relation is a correspondence between the
first set, called the domain, and a second set,
called the range, such that each member of
the domain corresponds to at least one
member of the range.
Domain: set of all values of x
Range: set of all values of y
Function
A function is a correspondence between a first set,
called the domain, and a second set, called the
range, such that each member of the domain
corresponds to exactly one member of the range.
Function
7
7
0
2
2
49
0
4
Not a Function
2
4
5
7
6
3
8
Function?
Which of the following relations is a function?
{(8, 2), (8, 4), (7, 3)}
{(6, 4), (1, 4), (7, 4)}
Notation for Functions
• The inputs (members of the domain) are values
of x substituted into the equation. The outputs
(members of the range) are the resulting values
of y.
• f(x) is read “f of x,” or “f at x,” or “the value of f
at x.”
• Example: Given f (x) = 3x2  4, find f (6).
Function
We are going to be dealing with
functions where we think of the . . .
function- as a rule that tells how to
determine the dependent variable (y)
for a specific value of the independent
variable (x).
Four Ways to Represent a Function
• Verbally (description in words)
• Numerically (by a table of values)
• Visually (by a graph)
• Algebraically (by a formula)
A Function Verbally
ex.
The amount of sales tax depends on the
amount of the purchase.
A Function Numerically
Use a table of values:
x
3
0
-1
y
19
10
7
A Function Visually
Any function can be visually represented
by a graph.
y
Dependent variable
x
Independent variable
A Function Algebraically
f(x) = 3x + 10
To solve a function: Evaluate the function.
Function Notation
“y is a function of x”
y depends on x
y = f (x)
f (x) is just another name for y !
ex. f(x) = 3x + 10
Vertical Line Test
• If every vertical line intersects a graph in
no more than one point, then the graph is
the graph of a function.
Example: Does the graph represent a
function?
The graph is a function
because we cannot find a
vertical line that crosses
the graph more than once.
Example: Does the graph represent a
function?
The graph is not a
function. We can find a
vertical line that
crosses the graph in
more than one point.
Finding Domains of Functions
Find the indicated function values and determine whether
the given values are in the domain of the function.
1
f (1) and f (5), for f ( x) 
x 5
f(1) =
Since f(1) is defined, 1 is in the domain of f.
f(5) =
Since division by 0 is not defined, the number 5 is not in the
domain of f.
Another Example
Find the domain of the function
3x  10 x  8
g ( x)  2
x  3x  28
2
Solution:
We can substitute any real number in the numerator, but
we must avoid inputs that make the denominator 0.
Solve x2  3x  28 = 0.
(x  7)(x + 4) = 0
x  7 = 0 or x + 4 = 0
x = 7 or
x = 4
The domain consists of the set of all real numbers
except 4 and 7 or {x|x  4 and x  7}.
Visualizing Domain and Range
Keep the following in mind regarding the graph
of a function:
• Domain = the set of a function’s inputs,
found on the horizontal axis.
• Range = the set of a function’s outputs,
found on the vertical axis.
Domain and Range
(How to find domain and range graphically)
Domain
The set of a function’s inputs, x-values,
found on the horizontal axis
Range
The set of a function’s outputs, y-values,
found on the vertical axis
Example
Graph the function.
Then estimate the
domain and range.
f ( x)  x  1
f ( x)  x  1
When to restrict Domain?
(How to find domain algebraically)
1. Fractions
–
You can’t have zero in the denominator!
2. Even root radical
–
You can’t have a negative under an even root
radical.