Understanding the Reciprocal Function: Definition, Graph, and Properties

reciprocal function

f(x) = 1/x

The reciprocal function is a mathematical function that produces the reciprocal value of a given input. The reciprocal of a number x is defined as 1 divided by x. Symbolically, the reciprocal function can be represented as f(x) = 1/x.

The reciprocal function is defined for every non-zero number except for zero itself since division by zero is undefined. As the input values approach zero from the positive side, the reciprocal function increases (tends to positive infinity). Similarly, as the input values approach zero from the negative side, the reciprocal function decreases (tends to negative infinity). This behavior is because smaller non-zero numbers have larger reciprocals.

The graph of the reciprocal function is a hyperbola, with vertical and horizontal asymptotes. The vertical asymptote occurs at x = 0, since the reciprocal function is undefined at x = 0. The horizontal asymptote occurs at y = 0, meaning that the reciprocal function gets closer and closer to zero as the input values increase or decrease without bound.

One important property of the reciprocal function is that it maps values between -1 and 1 onto real numbers larger than 1 or smaller than -1. For example, if we take the reciprocal of 0.5, we get 1/0.5 = 2. If we take the reciprocal of -0.2, we get 1/(-0.2) = -5. This property helps in understanding the behavior of the reciprocal function and its application in various mathematical problems.

Overall, the reciprocal function is a fundamental mathematical concept that is used in various areas of mathematics and science, such as in calculus, algebra, physics, and engineering.

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