Analyzing the Equation y = 1/x: Domain, Range, Graph, and Asymptotes

y= 1/x

To understand the equation y = 1/x, it is helpful to have some prior knowledge of algebra and graphing

To understand the equation y = 1/x, it is helpful to have some prior knowledge of algebra and graphing. This equation represents a mathematical relationship where the value of y is inversely proportional to the value of x. In other words, as x increases, y decreases, and vice versa.

Let’s analyze different aspects of this equation:

1. Domain and Range:
The domain of this equation is all real numbers except x = 0. Since division by zero is undefined, x cannot take the value of zero. Thus, the range of the equation is also all real numbers except y = 0.

2. Graph:
To graph the equation y = 1/x, we can create a table of values. Choose a few values for x and calculate the corresponding values of y.

For example:
x = -3, y = -1/3
x = -2, y = -1/2
x = -1, y = -1
x = 1, y = 1
x = 2, y = 1/2
x = 3, y = 1/3

Plotting these points on a graph, we get a hyperbola that passes through the points (-3, -1/3),(-2, -1/2), (-1, -1), (1, 1), (2, 1/2), and (3, 1/3).

The graph will have two branches, one in the first quadrant and the fourth quadrant, and the other in the second quadrant and the third quadrant. There will be asymptotes along the x-axis and y-axis due to the restrictions of division by zero.

3. Asymptotes:
The equation y = 1/x has two asymptotes: the vertical asymptote (x-axis) at x = 0 and the horizontal asymptote (y-axis) at y = 0. These asymptotes represent the values that the function approaches but never actually reaches as x or y approaches infinity or negative infinity.

In conclusion, the equation y = 1/x represents an inverse relationship between x and y. As x gets larger or smaller, y approaches zero. The graph of this equation is a hyperbola with asymptotes at x = 0 and y = 0. It is important to note the domain and range restrictions of this equation, which exclude x = 0 and y = 0, respectively.

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