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Algebraic functions of a real variable

Summary

This lesson introduces the concept of function, with its domain and its range, and studies the algebraic functions of a real variable: those obtained from the variable by means of sums, differences, products, quotients, powers and roots. It teaches how to determine the domain of a function defined by a formula and how to compute the range in simple cases, studies the graphs of powers and of roots, and classifies functions into algebraic and transcendental, and algebraic functions into polynomial, rational and irrational.

Learning objectives

  1. State the definition of function and distinguish, in each case, its domain, its codomain and its range.
  2. Determine the domain of an algebraic function defined by a formula.
  3. Compute the range of simple algebraic functions by solving for the variable.
  4. Sketch the graphs of powers with natural exponents and of roots from their monotonicity and their parity.
  5. Classify a function as polynomial, rational, irrational or transcendental.

Variables and algebraic expressions

A variable is a symbol that stands for a place where a number can be written. For real numbers the letters xx, yy and tt are traditionally used; the letter zz is usually reserved for complex numbers. When many variables are needed, subscripts are used: x1,x2,…,xnx_1, x_2, \ldots, x_n.

An algebraic expression is one formed from variables and real numbers by means of a finite number of sums, differences, products, quotients, powers with integer exponents and roots. For example:

x3+5x+6x,2x+3x−13x^3 + 5x + \frac{6}{\sqrt{x}}, \qquad \frac{2x + 3}{\sqrt[3]{x - 1}}

An algebraic expression does not always make sense for every value of the variable: the first one, for example, does not if xx is negative, because the square root does not exist, nor if xx is zero, because one would be dividing by zero. Determining precisely where it makes sense is the first problem raised by the study of functions.