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Relations and Functions
Definitions:
Relation: a set of ordered pairs
Domain: the first number in an ordered pair (usually x, or input, or independent variable)
Range: the second number in an ordered pair (usually y, or output, or dependent variable)
Function: a relation in which every input is assigned exactly one output. (every x value
has only one y value.)
You can check to see if a relation is a function by mapping out the domain and range and
seeing if every domain has only one range. If a domain is assigned more than one range,
then it is not a function. You can also check by graphing the points. You then use the
vertical line test to see if the graph crosses the vertical line more than once. If the vertical
line crosses the graph more than once, it is not a function.
Function notation is when you use f(x), g(x), or h(x) to represent the outputs of a function.
f(x) is read “f of x” or “f is a function of x.” This does not mean that f is a variable or that
( ) means multiplication.
To find a range from a given domain: take the values given for the domain and plug them
into the function for x and get the resulting output for the range. You can make a table if it
is easier. Below is an example of making a table of values:
Make a table for f(n) = -2n2 + 7. Use 1, 2, 3, and 4 as domain values.
n
1
2
3
4
-2n2 + 7
-2(1)2 +7
-2(2)2 +7
-2(3)2 +7
-2(4)2 +7
f(n)
5
-1
-11
-25
Here is an example of just finding the range without a table:
Find the range of the function f(a) = -3a + 5 for the domain {-3, 1, 4}
-3(-3) + 5 = 9 + 5 = 14
-3(1) + 5 = -3 + 5 = 2
-3(4) + 5 = -12 + 5 = -7
so, the range is {-7, 2, 14}
Examples:
Determine if the following relations are functions:
1.
{(3, 7), (3, 8), (3, -2), (3, 4), (3, 1)}
2.
{(2, 5), (3, -5), (4, 5), (5, -5)}
Make a table for each function using the domain of {1, 2, 3, 4}
3.
f(x) = x + 7
x
x+7
4.
f(x)
x
y = x2 - 4
x2 - 4
y
Find the range of the following functions for the domain {-2, 0, 5}
5.
f(x) = x – 6
6.
y = -4x
7.
f(x) = x2 + 1
Functions Rules, Tables and Graphs
Be able to model function rules with tables and graphs. To graph linear functions, choose
3 inputs that are easy to find outputs for (usually 0, 1, & 2). If there is a fraction as a
coefficient of x, choose inputs that are multiples of the denominator. To graph non-linear
functions, use -2, -1, 0, 1, 2 as your inputs. Non-linear functions are either absolute value
or the variable x is squared.
Model each rule with a table of values and graph.
8.
f (x)  3x  2

y
9.
1
x5
2
10.
f (x)  x 2 1


Slope and Rate of Change
Remember that slope and rate of change are the same. It is the vertical displacement over
the horizontal displacement, or the change in the dependent variable over the change in the
independent variable, or the change in y over the change in x. The formula for slope is
y  y1
m 2
x 2  x1
Find the slope of the line that passes through the following points.
11.
(3, -4) and (-2, -5)

12.
(7, -3) and (5, 6)
Slope-intercept
Remember that the slope-intercept form of the equation is y = mx + b, where m is the slope
and b is the y-intercept.
Find the slope and y-intercept of the following, then graph each line.
2
13. y  2x  3
14. y  x  2
5
slope: __________

y-intercept:__________
15.
slope: __________

3x  2y  8
16.
slope: __________

y-intercept:__________
y-intercept:__________
3y  2x  9
slope: __________

y-intercept:__________
Graphing with intercepts
When graphing using intercepts, the equation will be in standard form (Ax + By = C). You
find the x-intercept by setting y = 0 zero and solving for x. You find the y-intercept by
setting x = 0 and solving for y.
Graph the following using intercepts.
17.
4x  3y 12

18.
2x  y  6
20.
x7

19.
y  6
slope: __________
slope: __________
y-intercept:__________
y-intercept:__________
Direct Variation
When the value of one variable in an equation increases (or decreases) as another variable
in the same equation increases (or decreases), the variables are said to vary directly
An equation in the form y  kx , where k is a constant, expresses a direct variation. k is
y
called the constant of variation and is found using the equation, k  .
x
Some examples of situations where one quantity varies directly with another are:

•
Gasoline
and cost
•
Distance and time traveled

•
Circumference and radius of a circle
The graph of a direct variation equation is a line through the origin with a slope equal to
the constant of variation.
Examples
Write the direct variation equation that includes the given point.
21.
(3, -9)
23.
The distance (D) traveled varies directly with the time (t) it takes to travel that
distance at a constant speed. Suppose you travel 120 miles in 2 hours. How far
would you travel at this same rate for 5 hours
24.
A person’s paycheck (p) varies directly as the number of hours (h) worked. For
working 15 hours, the pay is $78.75. Find the pay for 35 hours of work.
25.
The amount (A) of money a family gives to charity varies directly as its income (I).
Last year, the family earned $72,350 and gave $6,362 to charity. How much will
they give if they make $75,000 this year? (round to the nearest whole $)
26.
The cost (c), in dollars, of gasoline varies directly as the amount (a), in gallons
purchased. The cost of 17 gallons of gas is $42.33. What is the cost of 23 gallons
of gas?
Bonus: what is the cost per gallon of gas?
22.
(4, 2)