Understanding the Hyperbolic Relationship: Exploring the Equation y = 1/x and its Mathematical Properties

y = 1/x

The equation y = 1/x represents a mathematical relationship between two variables, y and x

The equation y = 1/x represents a mathematical relationship between two variables, y and x. In this equation, the variable x is in the denominator, which means that as x increases or decreases, y will change accordingly.

This equation represents a hyperbola, a type of curve with two branches that extend infinitely. The curve will never touch or cross the axes, as x cannot be equal to zero. The vertical asymptote occurs at x = 0, and the curve approaches but never reaches this value. The horizontal asymptote occurs at y = 0, since as x approaches positive or negative infinity, the value of y gets closer and closer to zero.

To understand this relationship better, we can look at a table of values or plot the graph of the equation. Let’s choose some values for x and calculate the corresponding values of y:

When x = -2, y = 1/(-2) = -1/2
When x = -1, y = 1/(-1) = -1
When x = 0, y = undefined (division by zero is not defined)
When x = 1, y = 1/1 = 1
When x = 2, y = 1/2

These values show that as x moves away from zero towards positive or negative infinity, y approaches zero. Similarly, as x moves towards zero from positive or negative infinity, y tends towards positive or negative infinity.

Let’s plot the graph of this equation:

In this graph, the curve opens to the right and left, with the asymptotes at x = 0 and y = 0. The curve approaches these asymptotes but never touches them.

In conclusion, the equation y = 1/x represents a hyperbolic relationship between two variables, where y is equal to the reciprocal of x. It is important to remember that division by zero is undefined in mathematics, so the value of y is undefined when x = 0.

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