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 $$x$$.