3.5
Introduction To Functions
Homework: Odds 1-5, 13-19, 23-29, 31-45
| Nancy J. McCormick's Home Page | DSPM 0800 Class Notes |
· Functions are used to calculate many important quantities.
For
each valid input, x, a function produces exactly one output, y, which
may be represented by an ordered pair
.
We say that y is a function of x because the
output y is determined by and depends on the input x. So y is called the dependent
variable and x is the independent
variable. To emphasize that y is a function of x, we use the notation:
.
Example: Tom is paid an hourly wage of $8. Use
a function to calculate the weekly wage Tom earns after working x hours.
Verbal representation: Multiply each hour worked, x, by 8 to obtain the weekly wage, f(x) or y, in dollars.
Numerical representation:
|
x (hours) |
0 |
5 |
10 |
15 |
20 |
25 |
|
f(x) (weekly wage in $) |
0 |
40 |
80 |
120 |
160 |
200 |
Graphical
representation: A graph visually associates an x-input with a y-output.
The ordered pairs from the table can be plotted to form a graph or use the
equation, y = 8x.
Use grapher: Enter the equation
in Y= window, y = 8x.
Change graphing window to [0, 50, 5, 0, 400, 40, 1] to view the graph.
GRAPH:
Draw the graph on your paper, labeling the increments on the x-axis and on the
y-axis. Remember the increments for x are by 5 and the increments for y are by
40, both axes beginning at 0.
Construct the TABLE, starting at 0 with increments of 5.
By the table, find what weekly wage Tom will receive after he works 40
hours? $320
Example: (Source: Rockswold, G. (2002), Intermediate
Algebra through Modeling and Visualization.)
People who sustain leg injuries often require crutches. A proper crutch length
can be estimated without using trial and error. The function given by
can compute an appropriate crutch length in inches for a person x inches
tall. (Source: Journal of the American Physical Therapy Association).
a) Graph the function in the viewing window [60, 72, 2, 40, 60, 5, 1].
Interpret the graph.
The line has positive slope, so as the x-values
increase, the y-values increase. The graph indicates that taller people need
longer crutches.
b) Construct the table starting at 60 with increments of 2. What is the proper
crutch length for a person 6 feet tall? Convert feet to inches: 6
feet = 6(12 inches per foot) = 72 inches.
A person 6 feet tall needs crutches that are about 54
inches long.
· Definition
of a function:
A function f is a set of ordered pairs
(x, y) where each x-value corresponds to exactly one y-value.
· Identifying a Function.
1. From a diagram, a list of ordered pairs, or a table: Each input must
correspond to exactly one output.
Example:
|
x |
1 |
2 |
3 |
1 |
4 |
|
y |
-4 |
8 |
2 |
5 |
-6 |
This table does not represent
a function because input x = 1 produces two outputs, -4 and 5.
2. From a graph: Use the Vertical Line Test. Can a vertical line be drawn
anywhere on the graph that intersects the graph more than once? If so, the
graph does not represent a function.
Example:
This graph does not represent a function because a
vertical line can be drawn, as shown below, that intersects the graph at more
than one point.
· Evaluating algebraic
representations of a function.
Find the function values:
.
Substitute the given values in place of x.
Find:
.
Given the f(x) value or y-value, substitute in the
place of f(x), then solve for x.
·
Reading a graph to evaluate functions.
Given the graph of a function, f.
1. Find f(3).
Go to 3 on the x-axis, move vertically to the point on
the graph with x-value of 3, then move horizontally to the y-axis and
determine the y-value is 3. So f(3) = 3.
2. Find x such that f(x) = 1.
We are given the y-value of 1, go to 1 on the y-axis,
move horizontally to the point on the graph with y-value of 1, then move
vertically to the x-axis to determine the x-value is 1. So x = 1.