A truth value is either true or false; any statement with no variables in it should evaluate to true or false.
A constant is, in this class, usually just a real number. Any expression with no variables should evaluate to a constant, but we use one dimension to graph a constant on a number line.
A set is, in this class, a set of real numbers. A statement with one variable defines a set, such as {x | x < 2}, the set of real numbers that are less than 2. We again use one dimension to graph a set.
A function is a rule for taking one number (the input, or argument) and using it to calculate a number (the output, or value). An example is (x → x − 2), the rule which subtracts 2 from any number. To graph a function, we need two dimensions, one for the input and one for the output.
A relation is a set of ordered pairs instead of a set of individual numbers. An example is {x,y | x + y < 2}.
We can continue with binary functions, ternary functions, etc, which take two or more numbers as inputs; and we can continue with ternary relations, quaternary relations, etc, which relate three or more numbers. But we will not actually study these in this class.
One of the basic principles of the theory of functions is that the only information necessary to specify a function is to show how to calculate its value at any argument. Therefore, if I write
f(x) = x − 2and state that this holds for every real number x, then I have completely specified the function f. Usually in class (and always in the book), we will define a function in this way. Really it's just another way of saying that f is the function (x → x − 2). It's a handy method, because you have an equation in which you can replace x with any other expression (since the equation holds for every real number x). For example, using the function f above, we can replace x with 5 to calculate that
f(5) = (5) − 2 = 3.Or we can say that
f(2x + 3) = (2x + 3) − 2 = 2x + 1.Notice that I always put parentheses around an expression when I substitute it for a variable; in this case, it wasn't really necessary, but it's best to play it safe.
Sometimes a vertical line doesn't go through the graph at all! This happens when f(x) is undefined. The domain of f is the set of all inputs where f is defined:
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