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: $y=x^2$ Find $y$ when $x=5$ . Answer: $y=5^2=25$

Example, not a function: $y^2=x$ . Find $y$ when $x=25$ . Answer: $y=5$ or $y=-5$ – this relation is not a function. Explicitly, $y=\pm \sqrt{x}$ .

Interpretation 2: When looking at a graph, if you put ruler vertically on any $x$ 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 $y=x^2-3$ . We can express the same relation using function notation:

$f(x)=x^2-3$

A function, like a relation, is a set of ordered pairs. The first element in the ordered pair is the variable $x$ , the second element is the value of the expression, $x^2-3$ . Some ordered pairs generated by this function are:

$\{(-1,-2),(0,-3),(1,-2),(10,97),\dots\}$

Each function has:

a name, (in our example, $f$ );
one or more variables (in our example, $x$ );
an expression (in our example, $x^2-3$ ).

Evaluate: calculate the output

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

Example: $f(x)=x^2-3,\quad \{-5\leq x \leq 10\}$

Evaluate (a) $f(4)$ ; (b) $f(0)$ ; (c) $f(12)$

(a) $f(4)=4^2-3=16-3=13$ The ordered pair $(4, 13)$ is a member of this relation.

(b) $f(0)=0^2-3=0-3=-3$ Zero is usually an easy calculation! The ordered pair $(0,-3)$ is a member of this relation.

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 $x$ :

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

Not Well Defined

Consider the function:

$f(x)=\frac{1}{x-2}$

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, $x=2$ . Better to say:

$f(x)=\frac{1}{x-2}$ , $x$ is a real number but $x \ne 2$ .

or more simply,

$f(x)=\frac{1}{x-2}$ , $x \ne 2$

Practice finding the input

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

$f(x)=4x-1$ Find $x$ when $f(x)=19$ .

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

$\begin{align*}4x-1&=19 \\[10pt] 4x&=20 \\[10pt] x&=5\end{align*}$

Your turn:

Multiple inputs for the same output is OK

In all of the functions in this last applet, there is only one input for each output – only one solution. However, a function can still be well defined when there are multiple inputs for the same output. An example would be the houses at Hogwarts. Let the domain be the set of students at Hogwarts, and the range is the set of houses (Gryffindor, Hufflepuff, Ravenclaw and Slytherin). Now every student is assigned one and only one house – this relation is a well defined function. So when asked ‘which house is Hermione in?’ The one and only answer is ‘Gryffindor’. However, when asked, ‘which student is in Gryffindor?’ there are many answers. This is an example of a ‘many to one’ relation.

All of the above, on mathisfun!

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Functions and Function Notation