1.5 Inverse Function Existence

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Explain in words why a given function does not have an inverse.

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What Is an Inverse Function?

Think of a function like a machine. You put a number in, it does something to it, and gives you a result out.

An inverse function is a second machine that undoes exactly what the first machine did. If the first machine turns 3 into 7, the inverse machine takes 7 and gives you back 3.

  • The inverse of a function called f(x) is written as f⁻¹(x).
  • The domain (the set of inputs) of f⁻¹(x) is the same as the range (the set of outputs) of f(x).
  • The range of f⁻¹(x) is the same as the domain of f(x).

💡 In simple terms: the domain and range swap between a function and its inverse.

Not Every Function Has an Inverse

This is the most important idea in this subtopic.

A function only has an inverse if it is a one-to-one mapping (also called a one-one function).

Let's break down what that means.

One-to-One vs Many-to-One Functions

✅ One-to-One Function (Has an Inverse)

A function is one-to-one if every single output value comes from exactly one input value.

  • In other words: no two different inputs give the same output.
  • Example: f(x) = 5x − 2
    • f(1) = 3, f(2) = 8, f(3) = 13 → every output is unique.
    • This function does have an inverse.

❌ Many-to-One Function (Does NOT Have an Inverse)

A function is many-to-one if two or more different inputs give the same output.

  • Example: f(x) = x² for x ∈ ℝ (all real numbers)
    • f(3) = 9 and f(−3) = 9
    • Two different inputs (3 and −3) give the same output (9).
    • This function does NOT have an inverse.

Why Does a Many-to-One Function Fail to Have an Inverse?

Here is the key reasoning you need to be able to explain:

If a function sends two different inputs to the same output, then the inverse would need to send that one output back to two different inputs at the same time.

But that is impossible for a function — a function must give exactly one output for each input. If the inverse tried to map one value to two different values, it would no longer be a function at all.

📌 The Rule: A function f(x) has an inverse if and only if f(x) is a one-to-one mapping. If f(x) is many-to-one, it does not have an inverse.

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