# Function NotationInstructor: Dr.Jo Steig

DEFINITION: A function can be defined in a variety of ways. Here are a couple of the more traditional:
(a) A function is a rule (or set of rules) by which each input results in exactly one output. The set of all acceptable inputs in called the domain. The set of all resulting outputs is called the range.

(b) A function is a set of ordered pairs in which each first component is matched up with exactly one second component.

For specific information about finding the domain of a function, see the Supplementary Reading entitled Domain

What both of these definitions has in common is that each input to the function can result in EXACTLY one output. The usefulness of this idea is that the function has predictable, deterministic outcomes.

EXAMPLES OF FUNCTIONS:

(a) The amount of your take home pay.

(b) The amount of your electric bill

(c) Mixing paint to match a particular color

(d) Making cookies from a recipe

(e) The amount of weight an elevator can lift

Because the concept of a function is used so often, a shorthand has been developed...we call it functional notation. (1) the name of the function is g

(2) the letter, x, stands for the input.

(3) the output (based on an input of x) is referred to as g(x)

(4) what the function does (its process) is to take the input, multiply it by 3, add 4 , then divide the result by the original input

To indicate that we want to put the number 5 into the function g and calculate the result we would write the following: In other words, when we put the number 5 into the function called g, the output is 19/5.

In the function g, the letter x is actually being used as a place holder. To illustrate this point, we can remove the x. This leads to the following: We indicate that something is going into the function by putting it into the parentheses.

So, g(x) represents the output of the function when x represents the input.

And g(5) represents the output of the function when 5 is the input,

and g(a + b) represents the output of the function when a + b is the input,

Here are some examples of functions. In each case notice the function has a name, a letter representing the input, and a specific process by which the output is determined. The name of the function is f, the input is represented by the letter x, and the output (when x is the input) is denoted by f(x). f( -3) represents the output of the function when - 3 is the input.
The process that is described by the function is described by the algebraic expression (x3 - 2x + 1)/(x + 1). The name of the function is g, the input is represented by the letter t, and the output (when t is the input) is denoted by g(t). g(a+b) represents the output when a + b is the input to the function.
See if you can describe the process that this function represents? The name of the function is h, the input is represented by the letter x, and the output (when x is the input) is denoted by h(x). h(t + 1) represents the output to the function when t+1 has been the input.

In the next example, the input to the function is the expression x + h PARENTHESES PLACEMENT: In Example 4 the input was the expression, x + h. That means that x + h goes into the function. We say that the function operates on the input x + h. This is VERY different from the expression f(x) + h, which means just add h to the function.

Again, look closely at the two different statements: When working with functions it is important to notice where the parentheses are placed as they change the meaning of the expression.

In the next example we will subtract two functions, simplifying where possible. In the last example we make use of a particular combination of functions that includes subtraction and division. It is called the difference quotient. The problem from EXAMPLE 5 forms the numerator of the quotient. You should try to repeat the steps that I have outlined here. In a effort to save space, I have not shown all of the steps that you might require to work this problem yourself. See if you can put in the intermediate steps so that you understand how to move from one step to the next. 