Understanding the Reciprocal Function: Properties, Examples, and Graph

the reciprocal function

The reciprocal function is a mathematical function that maps every non-zero number to its reciprocal, which is the number obtained by dividing 1 by that number

The reciprocal function is a mathematical function that maps every non-zero number to its reciprocal, which is the number obtained by dividing 1 by that number. The reciprocal function is often denoted as f(x) = 1/x.

To understand the reciprocal function, let’s consider some examples:

1. For any positive number x, the reciprocal of x is given by 1/x. For instance, the reciprocal of 2 is 1/2, the reciprocal of 5 is 1/5, and so on.

2. For any negative number x, the reciprocal of x is also given by 1/x. For example, the reciprocal of -3 is 1/(-3) or -1/3, and the reciprocal of -7 is 1/(-7) or -1/7.

3. The reciprocal of 1 is simply 1/1 or 1.

It is important to note that the reciprocal of 0 is undefined since division by zero is undefined in mathematics.

The graph of the reciprocal function f(x) = 1/x is a hyperbola, with x and y axes as its asymptotes. The graph approaches the axes but never intersects them.

Some key properties of the reciprocal function include:

1. As x approaches positive infinity or negative infinity, the graph approaches the x-axis and y-axis but never intersects them.

2. The reciprocal of a number greater than 1 is always between 0 and 1, while the reciprocal of a number smaller than 1 is always greater than 1.

3. The graph of the reciprocal function is symmetric with respect to the origin. This means that if a point (x,y) lies on the graph, then the point (-x,-y) also lies on the graph.

4. The reciprocal function is not defined at x=0 since division by zero is undefined. As a result, there is a vertical asymptote at x=0.

In summary, the reciprocal function maps every non-zero number to its reciprocal. It is denoted as f(x) = 1/x and has various properties and a graph that resembles a hyperbola.

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