Transcript
Section 2.1Functions and Graphs
Relations
Domain: sitting, walking, aerobics, tennis, running, swimming
Range: 80,325,505,720,790
Do not list 505 twice.
Example
Find the domain and the range.
98.6, Felicia , 98.3,Gabriella , 99.1, Lakeshia
Functions
A relation in which each member of the domain corresponds to exactly one member of the range is a function. Notice that more than one element in the domain can correspond to the same element in the range. Aerobics and tennis both burn 505 calories per hour.
Is this a function? Does each member of the domain correspond to precisely one member of the range? This relation is not a function because there is a member of the domain that corresponds to two members of the range. 505 corresponds to aerobics and tennis.
Example
Determine whether each relation is a function?
1,8 , 2,9 , 3,10
2,3 , 2,4 , 2,5
3,6 , 4,6 , 5,6
Functions as Equations
2
Here is an equation that models paid vacation days each
year as a function of years working for the company.
y=-0.016x .93 8.5
The variable x represents years working for a company.
The variable y re
x
presents the average number of vacation days
each year. The variable y is a function of the variable x.
For each value of x, there is one and only one value of y.
The variable x is called the independent variable because
it can be assigned any value from the domain. Thus, x can
be assigned any positive integer representing the number
of years working for a company. The variable y is called the
dependent variable because its value depends on x. Paid
vacation days depend on years working for a company.
Not every set of ordered pairs defines a function.
Not all equations with the variables x and y define
a function. If an equation is solved for y and more
than one value of y can be obtained for a given x,
then the equation does not define y as a function of x.
So the equation is not a function.
Example
Determine whether each equation defines y as a function of x.
2
2 2
4 8
2 10
16
x y
x y
x y
Function Notation
The special notation f(x), read "f of x" or "f at x"
represents the value of the function at the number x.
If a function named f, and x represents the independent
variable, the notation f(x) corresponds
2
2
to the y-value for
a given x.
f(x)=-0.016x .93 8.5
This is read "f of x equals -0.016x .93 8.5"
x
x
2
We are evaluating the function at 10 when we
substitute 10 for x as we see below.
(10) -0.016 10 .93 10 8.5
What is the answer?
f
Graphing Calculator- evaluating a function
1
Press the VARS key. Move the cursor to
the right to Y-VARS. Press ENTER on 1.
Function. Press ENTER on Y .
Type (10) then ENTER. You will now see the
same answer that you saw on the previous
screen when you evaluated the equation at x=10.
l
2
Press the Y = key. Type in the equation
f(x)= - 0.016x .93 8.5
Quit this screen by pressing 2nd Mode (Quit).
x 2( ) .016 x,T, ,n x .93 x,T, ,n + .85
Example
Evaluate each of the following.
2
2
Find f(3) for f(x)=2x 4
Find f(-2) for f(x)=9-x
Example
Evaluate each of the following.
2
2
Find f(x+2) for f(x)=x 2 4 ?
Is this is same as f(x) + f(2) for f(x)=x 2 4
x
x
Example
Evaluate each of the following.
2
2
Find f(-x) for f(x)=x 2 4
Is this is same as -f(x) for f(x)=x 2 4?
x
x
Graphs of Functions
The graph of a function is the graph of its ordered pairs.
First find the ordered pairs, then graph the functions.
Graph the functions f(x)=-2x; g(x)=-2x+3
x f(x)=-2x (x,y) g(x)=-2x+3 (x,y)
-2 f(-2)=4 (-2,4) g(-2)=7 (-2,7)
-1 f(-1)=2 (-1,2) g(-1)=5 (-1,5)
0 f(0)=0 (0,0) g(0)=3 (0,3)
1 f(1)=-2 (1,-2) g(1)=1 (1,1)
2 f(2)=-4 (2,-4) g(2)=-1 (2,-1)
See the next slide.
x
y
x
y
f(x)g(x)
Example
Graph the following functions f(x)=3x-1 and g(x)=3x
x
y
The Vertical Line Test
x
y
The first graph is a function, the second is not.
x
y
x
y
Example
Use the vertical line test to identify graphs in which y is a function of x.
x
y
Example
Use the vertical line test to identify graphs in which y is a function of x.
x
y
x
y
Obtaining Information
from Graphs
You can obtain information about a function from its graph.
At the right or left of a graph you will find closed dots,
open dots or arrows.
A closed dot indicates that the graph does not extend
beyond this point, and the point belongs to the graph.
An open dot indicates that the graph does not
extend beyond this point and the point
does not belong to the graph.
An arrow indicates that the graph extends
indefinitely in the direction in
which the arrow points.
Example
Analyze the graph.2( ) 3 4
a. Is this a function?
b. Find f(4)
c. Find f(1)
d. For what value of x is f(x)=-4
f x x x
x
y
Identifying Domain and Range from a Function’s Graph
x
yIdentify the function's domain and range from the graph
Domain (-1,4]
Range [1,3)
Domain [3, )
Range [0, )
x
y
Example
Identify the Domain and Range from the graph.
x
y
Example
Identify the Domain and Range from the graph.
x
y
Example
Identify the Domain and Range from the graph.
x
y
Identifying Intercepts
from a Function’s Graph
We can identify x and y intercepts from a function's graph.
To find the x-intercepts, look for the points at which the graph
crosses the x axis. The y-intercepts are the points where the graph
crosses the y axis.
The zeros of a function, f, are the x values for which f(x)=0.
These are the x intercepts.
By definition of a function, for each value of x we can
have at most one value for y. What does this mean in terms
of intercepts? A function can have more than one x-intercept
but at most one y intercept.
Example
Find the x intercept(s). Find f(-4)
x
y
Example
Find the y intercept. Find f(2)
x
y
x
y
Example
Find the x and y intercepts. Find f(5).
(a)
(b)
(c)
(d)
Find f(7).
x
y
0
1
1
2
(a)
(b)
(c)
(d)
Find the Domain and Range.
D:(- , ) R:(-5,7]
D:(-5, ) R: (- , )
D:(- , ) R: [-5, )
D:[- , ] R: [-5, ]
x
y
(a)
(b)
(c)
(d)
22 3( ) Find f(-1)
7
xf x
1
7
57
1
71
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