Inverse Function - Explanation

An inverse function is a key concept in mathematics that allows for understanding the relationships between functions and their "reversals." This concept is particularly important as it broadens our understanding of functions and their applications.

DEFINITION

It is important to mention that a function has an inverse function only if it is bijective. This means it must be both:

• INJECTIVE (one-to-one): Each element in the domain maps to a unique element in the codomain (different inputs give different outputs).
• SURJECTIVE (onto): Every element in the codomain is mapped to by at least one element from the domain (all possible outputs are reached).

If a function f maps x to y (f(x) = y), its inverse function, denoted f^-1(x), maps y back to x (f^-1(y) = x).

PROPERTIES

• GRAPH SYMMETRY: The graph of an inverse function f^-1(x) is symmetrical to the graph of the function f(x) across the line y = x.
• DOMAIN AND RANGE: The domain of the inverse function f^-1(x) is the range of the original function f(x), and the range of f^-1(x) is the domain of f(x).
• INVERSE OF THE INVERSE: If you have a function f(x) and its inverse function f^-1(x), the inverse of f^-1(x) is the original function f(x). That is, (f^-1)^-1(x) = f(x).

EXAMPLE

For a better understanding, let's look at an example. Let the given function be f(x) = 2x + 3. This function is bijective, which means it has an inverse function.
To find it:

1. Replace f(x) with y: y = 2x + 3.
2. Swap x and y to represent the inverse relationship: x = 2y + 3.
3. Solve for y:
   x - 3 = 2y
   y = (x - 3) / 2
   Therefore, the inverse function is f^-1(x) = (x - 3) / 2.

CONCLUSION

Invertibility is essential for understanding how functions can be "undone" or "reversed." Understanding these functions opens the door to a deeper comprehension of mathematical structures and allows for a better grasp of various mathematical concepts, such as solving equations and transforming graphs. Inverse functions are fundamental in algebra, analysis, and numerous other areas of mathematics.