FunctionsDefinition of a Function (Paraphrased)A function is an expression that applies over a specified set of values and returns a new value based off what value was originally inserted. Functions come in many different forms, but there is one rule: a single input can NOT result in multiple outputs. If a strange numeric relation manages to provide multiple outputs, then it is not a function. When graphing a function, a vertical line should not be able to cross the function more than once.
Domain and RangeAny generic input value is called an
independent variable. It is so named as there is nothing to determine what the input will be: the input is independent of any special rules.
A set of all real, possible independent variables for which the function will produce outputs is called the
domain.
Any generic output value is called a
dependent variable. It is so named as the independent variable and the function together determine what the output will be: the output is dependent on the input value.
A set of all real, possible dependent variables for that function is called the
range.
Most functions will be written in a form that equates the dependent variable to some modification of the independent variable, such as in Y = X + 1.
To determine what input is necessary to receive a certain output, or vice versa, is known as
evaluating the function for a variable. For example, given the above function, what input is necessary to get a value of 5?
Y = X + 1
5 = X + 1
5 - 1 = X + 1 - 1
4 = X
The answer is 4. In order to get an output of 5, you have to put in the input 4.
When plotting a particular point from a graph, you write down the coordinate pair. The coordinate pair is the pair of values linked together by the rule. Given the earlier example, you know that an input of 4 provides an output of 5. You would write down the coordinate pair surrounded in parentheses, with the independent variable first and the dependent variable second, separated by a comma.
(4,5)
When describing the domain and range of a function, one generally uses a special notation. For some functions, they are described simply by listing all of the possible input and output values and enclosing each of the two sets in brackets, like this:
Input/Domain: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
Output/Range: [5, 5, 8, 0, 4, 2, 6, 7, 3, 1]
The function exists with those given values: if a number not in the domain is input, then no output will result.
For some functions, such as Y = X + 1, there are an infinite number of real, possible values. This is when shorthand notation becomes necessary. You write down the left and right boundaries of the set (or infinity if no such boundary exists), and enclose in the proper punctuation.
Domain: (-infinity, infinity)
Range: (-infinity, infinity)
Instead of "infinity," you would draw the symbol for infinity.
There are certain rules to determine whether a parenthesis or a bracket should be used.
- If the "value" is infinity or negative infinity, use a parenthesis.
- If the value is excluded from the set, but a value infinitesimally smaller or larger is included in the set, then use a parenthesis.
- If the value is included in the set, use a bracket.
For example, the function Y = X
2 has a domain of all real, possible numbers, but its range can only be non-negative (i.e. zero and/or positive). You would write the domain and range as follows:
Domain: (-infinity, infinity)
Range: [0, infinity)
In another example, Y = ln(X) will not produce a value if its input is non-positive (i.e. zero and/or negative), but will produce outputs for all positive values.
Domain: (0, infinity)
Range: (-infinity, infinity)
The input value can go infinitely close to zero, but it cannot be zero or anything smaller than zero. A parenthesis is used. There is no highest possible input value, so "infinity" with a parenthesis is used.
The output value has no real boundary, so "-infinity" and "infinity" are used with there parentheses.