(Section 6)
What is a function?
Is it an equation? Is it a picture? Is it a graph? Think
for a few minutes about what you think you know about a
function?
If a caterpillar is crawling around on a
piece of graph paper, as shown below. Your teacher asks you to
determine the location of the creature on the graph paper at particular
times. Would you define the position of this caterpillar as a function
of time?**
As we learn ideas in math class
we develop concept images in our minds. Right now you have a concept image
of what a function is. Sometimes these ideas are correct and some times
they need a little modification. As you study functions in calculus you
will want to make sure your definition is clear. Here are some questions
you should think about.
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Question: Is a function always
described by a single rule?
Answer:
No--it is possible to split a domain and use different rules for different
parts of the domain. These functions are call piecewise functions.
One common piecewise function is the graph of
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because it is made up of two
separate rays. One is the equation f(x) = x and the other one is f(x) =
-x.
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Question: Is the graph of a function
always continuous?
Answer:
No--it is possible that a function could have a point of
discontinuity. Think about the function
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This function has no value a x
= 2, but there is a function value at every other x value. This is not
an asymptotic function. It is actually the same as the function y = x+2,
but with a hole at the point where x = 2 and y = 4.
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Question: Does a function have to be
one-to-one? In other words, is the equation y = 3 a function?
Answer:
No--a function does not need to be one-to-one. A function can take on
the same y value for every x.
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Question: Is a set of values in a chart
a function?
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Name
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Age
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Steve
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5
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Suzanne
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8
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Scott
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12
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Answer: Yes --a set of values in a
chart like the one above is a function. The chart assigns a single value (age) to each
person.
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Question: Can a graph such as the
following be a function?
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Answer: This graph is a
function. For each x value only one y value has been assigned. Of
course for some x values the same y value has been assigned. This graph
is modeling the greatest integer
of x. 
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Question: Is a single point is
a function?
Yes--A function can be a single point. In other words, that x value has
a specific y value, but all other x values have no y value assigned to
them.
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So what is a function? A
function is defined as a set of point(s) such that each x value is assigned to
exactly one y value. So you can see that each of the ideas presented above
can help you clear up your idea of a function.
Now that you have cleared up
your thinking about a function, let's return to our initial example. What about the caterpillar?
This answer is a bit tricky. The catapillar's position can be thought of as a
point (x,y). Time is another variable. Is the
path of the caterpillar or position of the caterpillar, a function of time? Yes it is. For every t
value of time there is exactly one location for the caterpillar on the graph
paper, even though the location probably is named with a coordinate (x,y).
You won't experience these types of functions until much later in your study of
Calculus. It is not usually part of a first two semesters of Calculus.
But you should know it is an example of a function. You will also study
some other types of graphs throughout the second semester of Calculus that will
also challenge you to think carefully. You will be studying parametric
equations and polar equations.
Check Your Understanding: (Remember
to write down your answers.)
1. Tell whether each of the following is
or is not a function
2. Which of the following indicate
that y is a function of x?
a.
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| b. y = x/3 |
| c. xy = 8 |
d.
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e.
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f.
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**Some of them material on functions is based on What do Students Really
know about Functions? by Barbara Edwards and Karen Graham in the December,
2001 Mathematics Teacher)
This page was modified on
05/29/10
© Rahn, 2000
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