Inverse of a function

Inverse of a Function

In mathematics, the concept of an inverse function is fundamental in understanding the relationship between two sets of numbers or variables. An inverse function essentially reverses the effect of the original function. If a function f takes an element x from its domain and maps it to an element y in its codomain, then the inverse function f⁻¹ takes y and returns the original element x.

Definition

For a function f: A → B, where A and B are non-empty sets, an inverse function f⁻¹: B → A exists if f is bijective (one-to-one and onto). This means:

  • One-to-one: For every x₁ and x₂ in A, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).
  • Onto: For every y in B, there exists an x in A such that f(x) = y.

Properties

When a function f has an inverse f⁻¹, the following properties hold:

  1. f(f⁻¹(y)) = y for every y in B.
  2. f⁻¹(f(x)) = x for every x in A.
  3. The domain of f is the codomain of f⁻¹, and vice versa.
  4. The graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Finding the Inverse of a Function

To find the inverse of a function f(x), follow these steps:

  1. Write the function as an equation: y = f(x).
  2. Swap x and y: x = f(y).
  3. Solve the equation for y to get y = f⁻¹(x).

Table of Differences and Important Points

Property Function f Inverse Function f⁻¹
Definition Maps x to y Maps y back to x
Notation f(x) f⁻¹(x)
Domain Set A Set B
Codomain Set B Set A
Bijectivity Must be bijective Inverse exists by default
Graph Reflection Across the line y = x Original graph of f

Formulas

If f and f⁻¹ are inverses, then:

$$ f(f⁻¹(x)) = x \quad \text{and} \quad f⁻¹(f(x)) = x $$

Examples

Example 1: Linear Function

Let's find the inverse of the function f(x) = 2x + 3.

  1. Write as an equation: y = 2x + 3.
  2. Swap x and y: x = 2y + 3.
  3. Solve for y: y = \frac{x - 3}{2}.

The inverse function is f⁻¹(x) = \frac{x - 3}{2}.

Example 2: Quadratic Function

Consider the function f(x) = x^2, where x ≥ 0. This function is not bijective over all real numbers, but it is bijective when restricted to x ≥ 0.

  1. Write as an equation: y = x^2.
  2. Swap x and y: x = y^2.
  3. Solve for y: y = \sqrt{x} (since x ≥ 0).

The inverse function is f⁻¹(x) = \sqrt{x}.

Example 3: Non-Invertible Function

The function f(x) = x^2 without any domain restriction is not invertible because it is not one-to-one (e.g., f(-1) = f(1) = 1).

Conclusion

Understanding the inverse of a function is crucial for solving equations, understanding symmetry in graphs, and analyzing real-world problems. Not all functions have inverses, but when they do, the inverse provides a powerful tool for 'undoing' the action of the original function.