A mirror shows an inverse image, where left and right are interchanged.
A playground has many opportunities for children to explore another kind of inverse.
Mathematics has different kinds of inverses as well. One inverse that you know about is called the additive inverse, also known as the opposite, or negative. The opposite of three is negative three, and the opposite of negative three is three.
The opposite function embodies the nature of additive inverses as a function.
The graph of the opposite function is the straight line passing through the origin with a slope of negative one.
Another kind of inverse in mathematics is the multiplicative inverse, or reciprocal. Reciprocals can be written with an exponent of negative one or as a fraction.
The reciprocal of three is one-third, and the reciprocal of one-third is three. Also, the reciprocal of negative three is negative one-third, and the reciprocal of negative one-third is negative three. Note that computing the reciprocal of a number doesn’t change its sign.
Zero has an opposite, but does not have a reciprocal. Also, there is no number which has zero as a reciprocal.
The reciprocal function embodies the nature of multiplicative inverses as a function.
The graph of the reciprocal function is the focus of this simulation. As a number gets larger and larger, its reciprocal gets smaller and smaller. This means that the positive x-axis acts as a sort of barrier that the graph cannot cross. The positive x-axis is an asymptote. Likewise, the reciprocal of a very small number is very large, so the positive y-axis is also an asymptote. Since computing a reciprocal doesn’t change signs, the same is true for the negative x-axis and the negative y-axis. The x-axis is called a horizontal asymptote and the y-axis is a vertical asymptote. The asymptotes are both special cases of lines, and equations can be used to describe them. The horizontal asymptote is a horizontal line, with equation y equal zero, and the vertical asymptote is a vertical line, with equation x equal zero. When the reciprocal function undergoes a transformation, the asymptotes undergo the same transformation, even though they are not made from points on the function. In graphs, the asymptotes are usually depicted with dotted lines. The simulation allows you to explore how transformations on the reciprocal function. The interface works as in Lesson 8, transformations of Graphs, so be sure you are familiar with that lesson before proceeding. Good luck and have fun learning!