In this activity, we will be using functions to describe the relationship between two variables. Functions model situations involving two variables (sometimes more), where one variable (the output of the function) depends only on the other (the input of the function). That is, we can determine the value of the dependent variable (output) just by knowing the value of the independent variable (input) and the relationship between the two variables (the function).
Just as we use letters as variables representing numbers, we use letters to represent functions. In both cases, there are many different uses of these variables:
| We can use a variable to temporarily represent a definite, known number:
If x = 3, then x2 = 9 |
We can use a variable to temporarily represent a definite, known function:
If f is the squaring function (i.e. f(x) = x2), then f(3) = 9 |
| We can use a variable to temporarily represent a definite, unknown number:
If x is the number of particles in the universe, then x is on the order of (roughly) 1086 |
We can use a variable to temporarily represent a definite, unknown function:
If f tells us the height of a ball thrown up in the air as a function of time, then f is increasing for a while, then decreasing. |
| We can use a variable to temporarily represent an indefinite number of some kind:
If x is a positive number, then x = y2 for some y. |
We can use a variable to temporarily represent an indefinite function of some kind:
If f is a strictly increasing function, then f takes on no maximum value. |
| We can use a variable to temporarily represent a generic number:
For any number x, (x+1)2 = x2+2x+1 |
We can use a variable to temporarily represent a generic function:
If f is a function, and x is in the domain of f, then f(x) is in the range of f. |
Notice that there are many ways of specifying a function, just as there are many ways of specifying a number. The most common way of specifying a function is to specify it's output for a generic input. For instance, we could define a function by saying f(x) = x2 + 2x.
Caution: many sources (including me, when I'm not careful) will refer to the function f as f(x). That is, they will say something like "Let f(x) be an increasing function" as opposed to "Let f be an increasing function".
However it may help to think of the expression 'f(x)' as being a combination of the function f and the number x, just as we think of the expression 'x+1' as being a combination of the number x and the number 1, or 'x+y' as being a combination of the number x and the number y. In the case of each of these expressions, the result represents a number, and so we can use those expressions in situations where we expect to find numbers: