What is the form of a reciprocal function?

The reciprocal function is defined as a function that takes the form of ( y = \frac{k}{x} ), where ( k ) is a constant. This structure indicates that the output ( y ) is inversely proportional to the input ( x ).

In this function, as ( x ) increases, ( y ) decreases, and vice versa. This relationship creates a hyperbolic graph that has two branches, typically situated in the first and third quadrants if ( k ) is positive, and in the second and fourth quadrants if ( k ) is negative. The characteristics of a reciprocal function can be observed through its asymptotic behavior, where it approaches the x-axis and y-axis but never touches them.

Understanding this concept is crucial as reciprocal functions are foundational in various mathematical applications, particularly in algebra and calculus.