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Welcome to the intriguing world of real functions! This tutorial will guide you through the fundamental concepts and properties of real functions and how they apply in mathematics. You can also access our Math Tutorials and Math Calculators from the quick links below.
To apply the concepts covered in the tutorial below in this tutorial, you can make use of our suite of associated calculators for real functions provided below, each has a further detailed tutorial. These calculators allow you to perform operations, analyze properties, and visualize graphs of real functions.
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A real function is a rule that assigns a real number to each element in a subset of the real numbers. The set of all possible inputs is called the domain, and the set of resulting outputs is called the codomain. In mathematical notation, a real function f can be represented as
A function of the form f(x) = mx + b, where m and b are constants.
A function of the form f(x) = ax2 + bx + c, where a, b, and c are constants, and a ≠ 0.
A function that is a sum of monomials of the form anxn, where an are constants.
A function of the form f(x) = ax, where the base a is a positive real number.
A function of the form f(x) = loga(x), where the base a is a positive real number different from 1.
A function is injective (or one-to-one) if every element of the domain is mapped to a distinct element of the codomain.
A function is surjective (or onto) if every element of the codomain has a corresponding element in the domain.
A function is bijective if it is both injective and surjective. Bijective functions have an inverse function.
A function is continuous if, roughly speaking, its graph is a single unbroken curve with no gaps or jumps.