MATH 1143
Functions and Function Notation
Section 1.1
Representing Functions
A function is a rule that takes certain numbers as inputs and assigns to each input number
exactly one output number. The output is a function of the input. Input numbers and
output numbers are also called variables.
Functions can be represented in one of four ways, which we typically refer to as “The Rule
of Four””:
1. Words
Mathematical problems (especially mathematical modeling problems) are often presented using the written word. The challenge is typically to rewrite the problem using
mathematical symbols and numbers.
2. Tables
A set of numerical values (occasionally words such as state names might be used)
to indicate the behavior of the function. This is often the case with real-world data
representing economic growth, population, sports statistics, etc.
3. Graphs
A pictorial representation of the data modeling a behavior. Temperature fluctuations,
medical data (EKG), and bacterial growth are classic examples. To calculate the value
of a function using the graph, we simply construct a vertical line segment from the
desired value to the line. The length of the vertical segment constructed is the value
of the function.
4. Formulas
The algebraic representation of the function. Often, this is the most “comfortable” of
all forms. The formula allows us to evaluate the function at various locations without
graphing the function or amassing large volumes of data
Mathematical Models
A mathematical model is a function describing an actual situation.
Example 1 The sales tax on an item is 6%. Express the total cost, C, in terms of the
price of the item, P .
Solution We see the cost C is the output and the price P is the
input. So we have
Cost = Price + Tax on the Item
C = P + 0.06P
C = 1.06P
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Functions and Function Notation
MATH 1143
Section 1.1
Function Notation
Function notation is used to indicate that a quantity is a function of another quantity.
We might say that the number of molecules, M , in a gas is a function of the volume of the
gas, V . We can write this expression in the form
Number of Molecules = Volume of Gas
M = f (V )
This indicates that the number of molecules in the gas (the output value) is a function of the
volume of the gas (the input value). We might also say that the output value is dependent
on the input value. Thus, M is the dependent variable and V is the independent variable.
2
.
1+x
Solution Let’s evaluate the function and determine a few sets of ordered pairs:
Example 2 Let y =
If we treat y as a function of x, that is y = f (x), then we have
f (0) = 2,
f (1) = 1,
2
f (2) = ,
3
whereas x is a function of y, which is x = g (y), gives us
2
g (2) = 0,
g (1) = 1,
g
= 2,
3
f (3) =
1
2
1
g
=3
2
Is This a Function?
Not all quantities that are related represent a function. Determining if a function is modeled
is important step in working with data. If a function is represented by the data, then every
input will have exactly one output.
Example 3 Let t represent the time (in years) after the year 2000. And let w represent
the number of San Francisco ’49er victories:
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MATH 1143
Functions and Function Notation
Section 1.1
Solution Looking at the data, we see that w is a function of t. There are no repeated
t−values, so there is no repetition of output values for any input value.
However, t is not a function of w since w = 7 has three different values t = 3, 6 and 8.
The Graph of a Function
The vertical line test can be used to determine if a particular graph is associated with a
function. If there is a vertical line which intersects a graph in more than one point, then the
graph does not represent a function.
Example 4 Which of the graphs represent y as a function of x?
Solution Looking at the graphs, we have the following results:
1. Graph 1 represents a function since any vertical line crosses the graph exactly one time.
This indicates each input value has exactly one output value.
2. Graph 2 is not a function of x since a vertical line intersects the graph more than once.
For example, f (0) = 2 and f (0) = −2.
3. Graph 3 is a function of x since at the endpoints of each interval, the point is included
on one segment but not on the other.
Example 5 A woman drives from Aberdeen to Webster, South Dakota, going through
Groton on the way, traveling at a constant speed for the whole trip:
1. Sketch a graph of the woman’s distance from Webster as a function of time.
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MATH 1143
Functions and Function Notation
Section 1.1
2. Sketch a graph of the woman’s distance from Groton as a function of time.
Solution The results are as follows:
1. The function touches the t−axis when d = 0 since that is when the woman reaches
Webster. When t = 0, the woman is at Aberdeen (the farthest point from Webster).
The constant slope of the graph indicates that her speed is constant for the duration
of the trip.
2. The function crosses the t−axis when d = 0 since that is when the woman reaches
Groton. Notice: d is positive both when the woman is in Aberdeen and when the
woman is in Webster, since both cities lie a positive distance away from Groton.
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