Understanding Preimage in Mathematics: Explained with Examples

preimage

In mathematics, a preimage refers to the original set of values or points that are mapped or transformed to a new set of values or points through a function or transformation

In mathematics, a preimage refers to the original set of values or points that are mapped or transformed to a new set of values or points through a function or transformation.

Let’s consider a function f that maps points from one set (called the domain) to another set (called the codomain). If B is a subset of the codomain, then the preimage of B under f is the set of all points in the domain that get mapped to some point in B.

Formally, the preimage of a set B under f is denoted as f^-1(B). This is read as “f inverse of B”.

To understand this concept better, let’s look at an example:

Suppose we have a function f: R -> R defined by f(x) = x^2. Here, R represents the set of real numbers.

If we consider the set B = {4, 9}, the preimage of B under f would be the set of all real numbers x such that f(x) is either 4 or 9.

To find the preimage, we need to solve the equation f(x) = 4 and f(x) = 9.

For f(x) = 4, we have x^2 = 4. Taking the square root of both sides, we get x = ±2.

Similarly, for f(x) = 9, we have x^2 = 9. Taking the square root of both sides, we get x = ±3.

Therefore, the preimage of B = {4, 9} under f is the set {-3, -2, 2, 3}. In other words, these are the values in the domain that get mapped to either 4 or 9 under the function f(x) = x^2.

It is worth noting that the preimage of a set under a function is not always unique. In our example, the preimage of B could also be expressed as {-2, 2}, as the square of -3 and 3 is also 9. The important point is that all the elements in the preimage do get transformed to the corresponding elements in the set B under the given function.

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