An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root. Let me give an example. Suppose you have the function f and g where f is a linear function and g is a constant function. Let f(x)=x and g(x) = -3. We can create another linear function h by multiplying f and g that is h(x) = -3x. We can also create another linear function l where l = f – g, that is l(x) = x-3.
What about quadratic functions? A quadratic function (with real roots) is a product of two linear functions. So we can make a quadratic function n by multiplying f and l for example. That is, n(x) = f(x) x l(x) = x(x-3). And cubic function? A cubic function is a product of three linear functions or of a quadratic function and a linear function. And quartic function? Well, you must have figured it by now. This process of creating function by multiplying linear functions produces a family of functions called polynomial functions so called because its algebraic representation is a polynomial.
What kind of function is produced when you divide a function by a function in x? Using the function defined earlier, what is g÷f? g÷l? f÷l? Getting the quotient of two polynomial functions give us a new family of functions: p(x) = -3/x; q(x) = -3/(x-3); and, r(x) = x/(x-3). These expressions defining the functions will not simplify to polynomial expressions so they do not belong to the family of polynomial functions. They belong to what is called the family of rational functions so called because they are defined by rational expressions.
You can also raise a function to a fractional power, that is get the nth root of the function. For example we can have t(x)= x^0.5. That is t(x)=sqrt of x. I don’t know what this family of function is called. Maybe we can call then nth root functions.
These three families — polynomial functions, rational functions, and nth root functions, all belong to the family of algebraic functions. Functions that are not algebraic functions are called transcendental functions.
You may also want to read ideas for teaching functions.