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Preimage of a Function

Tool to calculate preimage(s) (antecedents) by a function for given value from the function expression.

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Preimage of a Function -

Tag(s) : Functions, Geometry

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# Preimage of a Function

## Preimage Calculator

 Global domain for the function Reals R Integers Z Complex C

### What is a preimage by a function? (Definition)

The preimage of a value $z$ by the function $f$ are all the values for which the function $f(x) = z$.

### How to find a preimage of a function?

From the definition of the function (without graph)

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 $f$, which has a known curve, is equivalent to find the abscissae of the intersection(s) of the curve with the ordinate line $y = a$.

### What is the difference between image and preimage?

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 $x$.

### How to find the preimage of a hash function?

Hash functions are combinations of nonlinear functions specifically created so that their antecedents are not easily computable. It is generally faster 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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Preimage of a Function on dCode.fr [online website], retrieved on 2024-07-15, https://www.dcode.fr/function-preimage

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