Download Pre-Calculus I 1.2 Graphs, Functions and Models A relation is a set

Pre-Calculus I
1.2 Graphs, Functions and Models
A relation is a set of ordered pairs of real numbers.
The domain, D, of a relation is the set of all first coordinates of
the ordered pairs in the relation (the xs).
The range, R, of a relation is the set of all second coordinates
of the ordered pairs in the relation (the ys).
In graphing relations, the horizontal axis is called the domain
axis and the vertical axis is called the range axis.
The domain and range of a relation can often be determined
from the graph of the relation.
**If the domain or range consists of a finite number of points,
use braces and set notation.
**If the domain or range consists of intervals of real numbers,
use interval (or inequality) notation.
Problem Type #1: State the domain and range of the relation.
EX: {(-1,1), (1,5), (0,3)}
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A function is a special kind of relation that pairs each element
of the domain with one and only one element of the range. (For
every x there is exactly one y.) A function is a correspondent
between a first set, domain, and a second set, range.
In a function no two ordered pairs have the same first
coordinate. That is, each first coordinate appears only once.
Although every function is by definition a relation, not every
relation is a function.
EX: Which of the following relations are functions?
(2,  8), (3, 0), (1, 5),
(2, 5), (3, 5), (1, 5),
(2, 5), (3, 0), (2, 0)
To determine whether or not the graph of a relation represents
a function, we apply the vertical line test which states that if any
vertical line intersects the graph of a relation in more than one
point, then the relation graphed is not a function.
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Problem Type #2: Is the relation a function?
EX:
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Problem Type #3: Function notation and evaluating
functions (finding range values) . . .
EXAMPLE:
to
, and
2 x  3 ”.
of
x
is equal
f
is the name of the function.
x
is representative of an element in the domain of
f (x)
f
f ( x)  2 x  3 is read “ f
.
is representative of an element of the range of
means the same as y .
2 x  3 is the function rule.
EVALUATE:
f
f (5) 
f ($) 
f (x  1) 
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Examples:
Graph and find the domain and range of the following functions
f(x) = x2 – 5
f(x) = x3 – x
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Find the domain and range of the following
f(x) =
Determine whether given points f(1) and f(3) are in the domain of
f(x) =
Restricted values occur when the denominator is equal to zero.
(Why? because a zero in the denominator makes a function
undefined). Also restricted values occur when the value under the
radical is less than zero (meaning negative) for even indexed
roots.
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a. g(x) =
b. h(x) =
c.
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To graph functions using a graphing calculator.
Step 1: Hit the “y=” button (purple) located under the screen on
the left
Step2: You will see
y1=
y2=
…
You can enter your equation now
For instance if we wanted to graph y=2x+3 then you would
enter 2x+3 on this screen.
Step 3: Hit enter
Step 4: Hit the “Graph” button (purple) located under the screen
on the right.
This step will graph the function for you.
Note if you cannot see your graph then your window settings are
not set correctly. You need to hit the “window” button (purple)
located under the window. You should have the x and y max be
10 and the x and y min be -10, the increment should be 1.
You can also evaluate function values after you have entered
your function into the “y1=”.
Let’s say your function is f(x) = 2x+3 and you have this saved in
“y1=” then you can determine f(30) by simply choosing “y1” hit
enter open parenthesis then 30 then close parenthesis then enter
and your calculator will calculate this for you. the answer it will
give you is 63.
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Review of the rectangular coordinate system – axes,
origin, quadrants, ordered pairs, coordinates, signs of
coordinates of points, etc.
Y-Axis
Vertical Axis
Quadrant II
Negative x-values
Positive y-values
Quadrant I
Positive x-values
Positive y-values
(-,+)
(+,+)
X-Axis
Horizontal Axis
Quadrant III
Negative x-values
Negative y-values
Quadrant IV
Positive x-values
Negative y-values
(-,-)
(+,-)
Plot means to show the location of a point on the rectangular
coordinate system.
Ordered Pair: ( x , y )
x: is the x-coordinate (move on the x-axis) 1st coordinate
y: is the y-coordinate (move on the y-axis) 2nd coordinate
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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.
- Closed dots mean that the graph does not extend beyond
this point, and the point belongs to the graph
- Open dots mean 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-
Using the above graph give the following:
a) Explain why represents the graph of a function
b) Use the graph to determine what is
c) For what categories of
3.5.
is the output value greater than
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Identifying Intercepts
Two distinct points determine a line.
The points on a line have coordinates that make the equation
of the line true.
To find the y-intercept of a line, let x=0.
To find the x-intercept of a line, let y=0.
Problem Type #1: Given the equation of a line, you should
be able to find the intercepts. Graph the lines too.
EX 1:
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EX 2.
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