Tool to calculate preimage(s) (antecedents) by a function. The preimage of a value z by the function f are all the values for which the function f(x)=z.

Preimage of a Function - dCode

Tag(s) : Functions, Geometry

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Tool to calculate preimage(s) (antecedents) by a function. The preimage of a value z by the function f are all the values for which the function f(x)=z.

**From the definition of the function**

Finding the preimage(s) of a value \( a \) by a function \( f \) is equivalent to solving equation \( f(x) = a \).

Example: Calculating the preimage of \( 1 \) by the function affine \( f(x) = 2x + 1 \) is to solve \( 2x + 1 = 1 \iff x = 0 \). So the fiber of \( 1 \) by \( f \) is \( 0 \)

Example: Find the preimage of \( 4 \) by the 2nd degree polynomial function \( g(x) = x^2 \). Solve the equation \( x^2 = 4 \ iff x \pm 2 \). So the preimages of \( 4 \) by \( g \) are \( -2 \) and \( 2 \)

**From the curve of the function**

Finding the preimage(s) of a value \( a \) by a function \( f \) whose curve is known is to find the abscissae of the intersection(s) of the curve with the ordinate line \( y = a \).

If a function \( f \) is such that \( f(x) = a \), the image of \( x \) by the function (\ f \) is \( a \), and the preimage of \( a \) by the function \( f \) is \( x \).

Hash functions are combinations of nonlinear functions specifically created so that their antecedents are not easily computable. It is generally simpler to calculate from the function f all the pairs (x, a) (or as much as possible, where the most probable) and to compare with the value of the antecedent sought.

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Source : https://www.dcode.fr/function-preimage

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