In mathematics, the reciprocal or multiplicative inverse of a number, n, is n−1 = 1n, because this satisfies the multiplicative identity: n⋅n−1 = nn = 1. For a rational number ab, the reciprocal is given by ba.
Following this definition, for a function fx, the reciprocal function is y = 1fx. If fx is a rational function of the form fx = hxgx, its reciprocal function will be y =gxhx.
Interesting Properties of Reciprocal Functions
The reciprocal function has vertical asymptotes wherever the original function has x-intercepts, and x-intercepts wherever the original function has vertical asymptotes. If x0 is an isolated root of the original function, that is, if fx0 = 0 and fx ≠ 0 for other values of x near x0, then the reciprocal function will approach ± infinity at these points, creating vertical asymptotes. Conversely, if a vertical asymptote occurs in the original function at x0, that is, its value approaches ± infinity as x approaches a given value x0, then the reciprocal function will have a root at x0.
Intervals, in which the original function is increasing, correspond with intervals in which the reciprocal function is decreasing. Meanwhile, intervals in which the original function is decreasing, correspond with intervals in which the reciprocal function is increasing. This occurs because as fx increases, y = 1fx must decrease. Similarly, as fx decreases, y =1fx must increase.
The original and reciprocal functions will intersect at the points where fx = 1 and fx = −1. This occurs because the reciprocal function will have the same value as the original, since y = 11 =1 and y = 1−1 = −1.
Type a function of x in the box below. Click "Enter" to display it in the plot. Click "Show Reciprocal Function" to display the reciprocal of your function.
Enter Function: fx =
Original Function, fx:
Reciprocal Function, y = 1fx:
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