Inversion can mean different things in different mathematical contexts. In the simplest sense, it is the opposite of a particular mathematical term. In the context of addition functions, the opposite, or inverse, of a number is the negative quantity of that number so the two terms, when added together, produce the quantity zero. In the multiplicative sense, it is the quantity such that the two terms when multiplied together will produce the quantity one.
The graph of an function or equation is a graph of every possible answer to that function. If a student graphs a function, and its inversion, then he will see the idea behind an inverse. Students can graph the inverse of a function, without using any equations. By selecting the major points of a function's graph, then charting a matching point exactly the same distance from the X and Y axis only on the other side of the graph, students will create a mirror image of a function: its inverse.
One of the earliest uses of inversions that students learn is how to divide fractions. Students can do this by inverting the second fraction in the equation (called the divisor) by which the first fraction in the equation (called the dividend) is being divided. For example, if the divisor was 4/5, then its inversion would be 5/4; when these two quantities are multiples together, the result is one. After changing the divisor to its inverse, then the student can solve the original division problem by using multiplication on the resulting equation.
By combining inversion and a matrix of numbers, students have a short cut for solving systems of linear equations -- multiple equations with multiple variables, where each variable in the different equations has the same value. To do this, students create two matrices: one containing all the prefix values to each variable, and another containing the values on the left hand side of the equals sign for each equation. By multiplying the inversion of the matrix containing the prefix values on all the variables by the matrix containing the terms on the left hand side of the equation, the student will be left with a matrix where each item is the value of a variable.