# Functions and Function Notation

Definition: A function is any relation (set of ordered pairs) whereby for each element in the domain, there is exactly one element in the range.

Interpretation 1: When a calculation is performed, if there is only ever one possible answer the calculation can be classified as a function.

Example, a function: Find when . Answer:

Example, not a function: . Find when . Answer: or – this relation is not a function. Explicitly, .

Interpretation 2: When looking at a graph, if you put ruler vertically on any value, and if it only every crosses a graph once, the graph represents a function. If it crosses more than once, the graph does not represent a function.

# Function Notation

Suppose we have a relation . We can express the same relation using function notation:

A function, like a relation, is a set of ordered pairs. The first element in the ordered pair is the variable , the second element is the value of the expression, . Some ordered pairs generated by this function are:

Each function has:

• a name, (in our example, );
• one or more variables (in our example, );
• an expression (in our example, ).

# Evaluate: calculate the output

We can define a function by stating an expression and a domain.

Example:

Evaluate (a) ;   (b) ;    (c)

(a) The ordered pair is a member of this relation.

(b) Zero is usually an easy calculation! The ordered pair is a member of this relation.

(c) Well, 12 is not in the domain. So there is no value for for this function.

If a domain is not stated, it is assumed that the domain includes all real numbers.

# Practice finding the output

The functions in the following applet are defined for all values of :

A function is well defined when for every input, there is one output.

# Not Well Defined

Consider the function:

We would say that this is not a well defined function. It is not well defined as there is a number that does not have an answer – that is, . Better to say:

, is a real number but .

or more simply,

# Practice finding the input

Sometimes we may have the output of a function and we are required to calculate the input:

Find when .

In this case, we set up an equation and solve it: