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Preimage of a Function

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.

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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

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.

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

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 $$y = a$$.

### 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 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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