Elementary function

The knowledge of elementary functions is a prerequisite for the study of calculus. These are functions of integer, rational, real or complex numbers. A list of the types of elementary functions follows.

• A power function consists of a single term, containing a numeric coefficient multiplied times a variable raised to a non-negative integer power. The exponent determines what is called the degree of the function. An exponent of zero yields a constant function, where only the coefficient is written. A first-degree power function is written without the one exponent.

• Polynomial functions are sums of power functions. A power function is considered a polynomial with one term. The degree of the polynomial is the degree of the term with highest degree. Sums, differences, products and integer powers of polynomials are all polynomials.

Power functions and polynomials are given names by their degree. The named polynomial functions include: constant (zero degree), linear (first degree), quadratic (second degree), cubic (third degree), quartic or biquadratic (fourth degree), and quintic (fifth degree). Functions with degree higher than five are not usually referred to by any other name.

• Rational functions are formed by dividing polynomials. These are normally written as fractions. Since dividing a polynomial by a constant gives a polynomial result, a polynomial can be considered a rational function. Sums, differences, products, integer powers and quotients of rational functions are all rational functions.

• Algebraic functions include rational functions and roots of rational functions. These can be square roots or any higher roots. Sums, differences, products, quotients, powers and roots are known as the basic algebraic operations. Using any of these on algebraic functions will always yield an algebraic function.

All functions of a variable that are not algebraic are called transcendental functions. A list of the types elementary transcendental functions follows.

• An exponential function has a constant as a base, and a variable expression as an exponent. While any constant can be used as a base, the exponential function has as a base, the irrational number e. A logarithmic function is the inverse of an exponential function. The natural logarithm function is the inverse of the exponential function. The notation for the natural log of x is ln x. The notation for the base ten log of x is log x. the notation for other base log functions is log b x.

• The trigonometric or circular functions are transcendental elementary functions. These are the sine and cosine, their inverses, and the other functions derived from them. The trig functions can be defined in terms of the exponential function.

• Hyperbolic functions and their inverses are analogous in some ways to the circular functions mentioned above. They are based on the unit hyperbola in a similar fashion as the trigonometric functions are based on a unit circle.

All of the elementary functions can be defined in terms of the natural logarithm function, using the basic algebraic operations and the functional inverse process.

There are transcendental functions that are not elementary functions. Most of these are defined based on calculus integrals that can not be solved using elementary functions.

References

G. H. Hardy, A Course Of Pure Mathematics, 10th ed., Cambridge University Press, 1908, 1952.