Algebraic Functions

A function is a rule, correspondence, or mapping $x \mapsto y$ that assigns to each real number $x$ (the input) to a certain set $D$ and one real number $y$ (the output). The set $D$ is called the domain of the function, and $y$ is the dependent variable, since its value depends on the value of $x$. Because $x$ can assign any value in the domain $D$, we refer to $x$ as the independent variable. The set of values assumed by $y$ as $x$ runs through all values in $D$ and is called the range of the function.

Most calculators have special keys for some of the more important functions such as $x \mapsto \sqrt x$ and $x \mapsto {x^2}$. The use of letters of the alphabet to designate functions is not restricted exclusively to calculating machines. Although any letters of the alphabet can be us designate functions, the letters $f$, $g$, and $h$ as well as $F$, $G$, and $H$ are most common (letters of the Greek alphabet are also used). For instance, if we wish to d the square-root function $x \mapsto \sqrt x$ by the letter $f$, we write $f:x \mapsto \sqrt x$.

If $f:x \mapsto y$ is a function, it is customary to write the value of y that corresponds to $x$ as $f\left( x \right)$, read as “$f$ of $x$.” In other words, $f\left( x \right)$ is the output produced when function  $f$ is applied to the input $x$.

For instance, if $f:x \mapsto \sqrt x$ is the square-root function, then $f\left( 4 \right) = \sqrt 4 = 2$, $f\left( {25} \right) = \sqrt {25} = 5$, $f\left( 2 \right) = \sqrt 2 \approx 1.414$ etc, and in general, for any nonnegative value of x, $f\left( x \right) = \sqrt x$. Note carefully that $f\left( x \right)$ does not mean $f$ times $x$.