How is the notation f(x) used to represent a function value?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In mathematics, the notation $f (x)$ —pronounced "f of x"—is used to represent the value of a function corresponding to a specific input, or independent variable, $x$ .

This notation defines a relationship where, for every assigned value of $x$ , a unique value is determined. In the equation $y = f (x)$ , the variable $x$ represents an element from the function's domain, while $f (x)$ represents the resulting element in the range. Because the function must produce a unique output for every input, $f (x)$ cannot have more than one value for the same $x$ .

While $f$ is the most common symbol, other letters such as $g$ or $P$ are frequently used (e.g., $g (x)$ or $P (x)$ ) to denote functions, particularly when the specific nature of the relationship is unspecified or unknown.1