Understanding Reciprocal Functions: Definition, Characteristics, and Graphing Techniques

reciprocal function

A reciprocal function is a type of rational function where the dependent variable (usually denoted as y) is equal to the reciprocal of the independent variable (usually denoted as x)

A reciprocal function is a type of rational function where the dependent variable (usually denoted as y) is equal to the reciprocal of the independent variable (usually denoted as x). In other words, the reciprocal function is defined as:

y = 1/x

The graph of a reciprocal function has some distinct characteristics. Firstly, the function is undefined when the independent variable (x) is equal to zero, since division by zero is undefined. Therefore, there is a vertical asymptote at x = 0.

Secondly, as the value of x approaches positive or negative infinity, the reciprocal function approaches zero. This creates two horizontal asymptotes, one at y = 0 and the other at y = 0.

The shape of the graph of a reciprocal function is a hyperbola. It is symmetric with respect to both the x-axis and the y-axis. The closer the value of x is to zero, the steeper the graph becomes.

To better understand how to graph a reciprocal function, let’s consider an example:

Graph the reciprocal function f(x) = 1/x.

Start by creating a table of values by choosing different x-values and calculating the corresponding y-values using the formula of the reciprocal function. Let’s select x-values of -3, -2, -1, 1, 2, and 3:

For x = -3, y = 1/(-3) = -1/3
For x = -2, y = 1/(-2) = -1/2
For x = -1, y = 1/(-1) = -1
For x = 1, y = 1/1 = 1
For x = 2, y = 1/2
For x = 3, y = 1/3

Now plot these points on a coordinate plane. You will notice that the points on the left side of the y-axis are mirrored on the right side due to the symmetry of the graph. Also, remember to include the vertical asymptote at x = 0 and the horizontal asymptote at y = 0.

The graph of f(x) = 1/x looks like a curve that starts from the positive y-axis, approaches the vertical asymptote at x = 0, and extends towards the negative y-axis.

Note that when working with reciprocal functions, it’s important to keep in mind certain values that cause division by zero, such as x = 0. These values are typically excluded from the domain of a reciprocal function.

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