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Understanding functions and relations is crucial for exploring more advanced topics in mathematics and their applications in various scientific and engineering domains. In this module, you'll learn about what functions and relations are and you can practice with the provided exercises. Relations A relation is a set of ordered pairs....
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Often the domain of a function will be restricted to a subset of R. This subset is called an interval, and the end points are a and b. Intervals Often the domain of a function will be restricted to a subset of the set of real numbers, \(\mathbb{R}.\) This subset...
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If f-1(x) is the inverse function of a one-to-one function f(x) then f-1(x) is the set of ordered pairs obtained by interchanging the first and second elements in each ordered pair. Definition of an inverse function If \(f^{-1}(x)\) is the inverse function of a one-to-one function \(f(x)\) then \(f^{-1}(x)\) is...
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The absolute value of a number x gives a measure of its size or magnitude regardless of whether it is positive or negative. If a number is plotted on a number line then its absolute value can be considered to be the distance from zero. Introduction The absolute value of...
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Functions which have different rules for each subset of the domain are called hybrid functions. Sometimes they are referred to as piecewise defined functions. An example of a hybrid function is: \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1. \end{cases} \end{align*}\] Note that this hybrid...
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A linear graph is the simplest way of representing data or a functional relationship. Consequently an understanding of linear graphs is an essential requirement for many courses in science, engineering and mathematics. The Cartesian plane The Cartesian plane is defined by a pair of mutually perpendicular coordinate axes. The horizontal...
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A quadratic graph is the graph of a quadratic function. These graphs have applications in a wide range of fields. Keep reading to learn about parabolas and how to sketch them, then test your knowledge with some exercises. A quadratic function has the form \(y=ax^{2}+bx+c\) where \(a\neq0\) . Parabolas The...
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The known graphs of some simple functions and relations can be used to sketch related, but more complicated functions. If you know the graph of a function, you can often transform it to a graph of a more complex but related function. A simple example is the graph of the...
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Functions and graphs provide a way to represent, analyse, and understand relationships between different quantities. If you are looking to improve your knowledge of this area, the linked resources are a great way to start. Plotting points on a graph Graphs usually have points or markers on them. The location...
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The hyperbolic functions are analogous to the circular (trigonometric) functions and are widely used in engineering, science and mathematics. This module introduces hyperbolic functions, their graphs and similarities to the circular functions. Whereas circular functions are defined on a unit circle, the hyperbolic functions are defined on a hyperbola. Hyperbolic...
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The differentiation and integration of hyperbolic functions allow us to understand how these functions change and accumulate values which teaches us the essential tools for analysing complex mathematical and real-world systems. Keep reading to learn about these functions and put your new knowledge to the test with some exercises. See...
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Providing a function is one to one, it is possible to find an inverse function. This module discusses inverse hyperbolic functions, which are used in advanced calculus for integration and in the solution of differential equations. See Hyperbolic functions for a list of definitions. Inverse hyperbolic functions may also be...
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Information about functions and graphs to improve your maths skills in these areas. HF1 Hyperbolic functions This module introduces hyperbolic functions, their graphs and similarities to the circular functions. HF2 Derivatives and integrals of hyperbolic functions The hyperbolic functions are widely used in engineering, science and mathematics. HF3 Inverse hyperbolic...
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Graphs usually have points or markers on them. The location of this point is given by an ordered pair. Watch this video to find out more....
