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Defining Functions, Domain, and Range

April 29th, 2009 |

Author: Merrin

Functions are one part of the basis for calculus. A function is a machine that takes one number as an input and gives another number as an output. An example of a function is

$f(x)=x^2 +2$

For an input of the number two, the output is six.

$f(2)=2^2 +2=6$

A function can also be designed to accept more than input, such as $latex f(x,y)&s=1$ for example, but for most of our work we will be dealing with a single input. The set of allowed inputs is called the domain. Sometimes there can be required restrictions on which inputs are not allowed, for example division by zero is not allowed. Sometimes there can be other restrictions on the domain just be choice of a certain definition. Each element of the domain can only generate one possible output. For example, if we had $latex f(1) = 1&s=1$ and $latex f(1) = -1&s=1$ then the object f is not a function. The output of a function must be reproducible. Same input in, same output out.

The set of all output possibilities corresponding to each element of the domain is called the range. If all the real numbers are the domain then $latex -\infty<x<\infty &s=1$. Infinity is not a number so it can’t be included in the inequality. Another way to represent inequalities specifying an interval is through interval notation. If the endpoint of the interval is included this is specified by a closed bracket on that end, [ or ]. For example if an interval was zero to two inclusive of the end points we could write $latex 0\le x\le 2&s=1$ as $latex x \in [0,2]&s=1$ as shorthand.

{../images/Intervals.pdf}

An array of possible intervals with closed and open ends as well as extensions to infinity or finite points

Each function has its own rule and features. We can define a second function as

$g(x)=\cfrac{1}{x-2}$

Notice that this function has a problem for an input of two. Division by zero is not defined. It is therefore understood that the domain of this function has that point removed $latex x\in(-\infty ,\infty )\backslash \{2\} &s=1$. The domain is sometimes inferred from the form of the function or can just be stated with the definition of the function. New functions can be created from combinations of other functions.

$\begin{array}{l} h_1(x) = {f(x)+g(x)} \\ h_2(x) = {f(x)-g(x)} \\ h_3(x) ={f(x)g(x)} \\ h_4(x) ={f(x)/g(x) \quad g(x)\ne 0} \end{array}$