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1.1 What is a function?

The function concept

Consider the formula for the area of a circle in terms of its radius: \(A=\pi r^2\)

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 \(9\pi\). Furthermore, the area will always be \(9\pi\) when the radius is 3. It will not sometimes be \(9\pi\) and sometimes something else.

Now consider the formula \(B^2=G\).

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 no more than 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 \(Q\).

Calculator examples

Scientific calculators contain function keys.

For example, there is a squaring function on many calculators. When one enters a number and presses the button for the squaring function (usually marked with the symbol \(\fbox{\(x^2\)}\) the number will be replaced by its square. Notice the importance of getting only one result 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 \(y=x^2\), where \(x\) is the input number and \(y\) is the output number.

There are other function keys on most calculators.

For example, there may be a key marked \(\fbox{\(\pm\)}\) which changes the sign of the input number. Notice that this function could be represented by the equation \(y=-x\), 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.

Functions

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.