Inverse Function

DEFINITION AND MEANING

An inverse function is a function that "reverses" the action of the original function. If a function 'f' maps an element 'x' to 'y', then its inverse function, denoted as f⁻¹, maps 'y' back to 'x'. The condition for an inverse function to exist is that the original function must be bijective, meaning it is both injective (different 'x' values produce different f(x) values) and surjective (every 'y' value in the codomain/range is reached).

Mathematically, we write: If f(x) = y, then f⁻¹(y) = x,
which also means:
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

CHARACTERISTICS OF AN INVERSE FUNCTION

• The domain of function 'f' becomes the range of function f⁻¹.
• The range of function 'f' becomes the domain of function f⁻¹.
• The graph of function f⁻¹ is a mirror image of the graph of function 'f' with respect to the line y = x.

PROCEDURE FOR DETERMINING THE INVERSE FUNCTION

1. Write f(x) as an equation: y = f(x).
2. Swap the roles of x and y: x = f(y).
3. Solve the resulting equation for y – this becomes f⁻¹(x).
4. Determine the new domain and range for f⁻¹(x).

EXAMPLE

Let the function be f(x) = 2x + 3.

• Step 1: y = 2x + 3
• Step 2: x = 2y + 3
• Step 3: x – 3 = 2y → y = (x – 3)/2
• Therefore: f⁻¹(x) = (x – 3)/2

Let's check:
f(f⁻¹(x)) = 2 * [(x – 3)/2] + 3 = x – 3 + 3 = x
f⁻¹(f(x)) = ((2x + 3) – 3)/2 = 2x/2 = x

INVERSE FUNCTIONS FOR SOME COMMON FUNCTIONS

• f(x) = x³ → f⁻¹(x) = ³√x (inverse exists on R - all real numbers)
• f(x) = x² → not invertible on R because it's not injective; it becomes invertible if the domain is restricted to [0, ∞)
• f(x) = eˣ → f⁻¹(x) = ln(x)

CONCLUSION

An inverse function expresses the reversibility of a mathematical mapping. Its definition is based on swapping the dependency between input and output. For it to exist, the original function must be bijective. The inverse function plays an important role in algebra, analysis, and in solving equations where we want to "undo" the operation of a function.undo" the operation of a function.