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(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.
because it is made up of two separate rays.  One is the equation f(x) = x and the other one is f(x) = -x.
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.  
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.    
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.  
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

a.  graph
b.  graph
d.  graph
e.  graph

 2.  Which of the following indicate that y is a function of x?

a.  equation
b.  y = x/3
c.  xy = 8
d.  equation
e.  equation
f.  equation


Table of Contents

Section 5

Answer Section

Section 7



**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 03/21/16

Rahn, 2000