Functions and Their Graphs
Read this section for an introduction to functions and their graphs. Work through practice problems 1-5.
Functions Defined by Equations
If the domain consists of a collection of real numbers (perhaps all real numbers) and the range is a collection of real numbers, then the function is called a numerical function. The rule for a numerical function can be given in
several ways, but it is usually written as a formula. If the rule for a numerical function, , is "the output is the input number multiplied by itself", then we could write the rule as . The use
of an "" to represent the input is simply a matter of convenience and custom. We could also represent the same function by , or .
For
the function f defined by , we have that , , and . Notice that the two different inputs, 3
and –2, both lead to the output of 6. That is allowable for a function. We can also evaluate f if the input contains variables. If we replace the "" with something else in the notation
"", then we must replace the "" with the same thing everywhere in the equation:
,
, and, in general, .
For more complicated expressions, we can just proceed step–by–step:
.
Practice 2: For the function defined by , evaluate , and .