1.1
What is a function?
Consider the
formula for the area of a circle in terms of its radius: 
For any given
value of r, there is only one corresponding value of A. Thus, for example, if the radius of the circle
is 3, its area will be 9π. Furthermore, the area will always be 9π when the radius is 3.
It will not sometimes be 9π and sometimes something else.
Now consider the
formula 
.
For any given
value of G (except 0), there will be two values of B. Thus, for example, if G = 4, then either B
is 2 or B is -2.
In the first
example, A has a unique value for any given value of r. In the second example, B does not
have a unique value for any given value of
G. We say that A
is a function of r, but B is not a function of G.
In general, in
order for a quantity y to be a function of the quantity x,
it must be the case that, for any given value of x, there is at most one
corresponding value of y.
Exercise
1.1.1
In the second
example just given, decide whether or not G is a function of B.
Exercise
1.1.2
Make up a
formula in which W is a function of p
Exercise
1.1.3
Make up a
formula in which Q is not a function of t. Verify that this is the case by finding a
particular value of t for which there is more than one corresponding
value of Q.
Scientific
calculators contain function keys.
For example,
there is a squaring function on most scientific calculators. When one enters a number and presses the
button for the squaring function (usually marked with the symbol 
) the number will be replaced by its square. Notice the importance of getting only one
‘answer’ when pressing the squaring function button. If we sometimes got one result and sometimes another, we could
conclude that the calculator was not working properly. For any given input, we should always get
the same output.
We could
represent the squaring function on a calculator with the equation 
, where x is the input number and y is the
output number.
There are many
other function keys on a scientific calculator.
For example,
there is a key, usually marked ‘±’ which changes the sign of the input number. Notice that this function could be
represented by the equation 
, where x is the input number and y is the
output number.
Exercise
1.1.4
Find at least
two other function keys commonly found on a scientific calculator. Describe, in words what the key does to the
input number. Find an equation in x
and y which represents what the function does to the input number, where
x represents the input number and y represents the output number.
There are many
kinds of functions commonly used in Calculus and in other areas of mathematics. In this tutorial, we will encounter many of
them.