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Invertible Functions and Finding the Inverse

A function f: A to B is invertible if there exists a function f⁻¹: B to A such that f⁻¹(f(x)) = x for every x in A and f(f⁻¹(y)) = y for every y in B. A function is invertible if and only if it is bijective.

In one line: An inverse needs two promises: every codomain value is reached, and reached only once.

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Understand it

  • An inverse function reverses the action of the original function.
  • It exists only when every output in the codomain comes from exactly one input.
  • One-one gives uniqueness, and onto gives existence.
  • Both are necessary.

Example

For f: R to R, f(x) = 5x - 7, set y = 5x - 7. Then x = (y + 7)/5. Hence f⁻¹(y) = (y + 7)/5, or f⁻¹(x) = (x + 7)/5.

Formula and rule

Invertibility theorem: f is invertible if and only if f is bijective.

Inverse identities: f⁻¹∘f = identity on A and f∘f⁻¹ = identity on B.

Method: write y = f(x), solve for x in terms of y, then replace y by x after stating the inverse rule and domain.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let f: R to R be f(x) = 2x + 5.

  1. 1First prove bijective.
  2. 2One-one: f(a) = f(b) gives 2a + 5 = 2b + 5, so a = b.
  3. 3Onto: for any y in R, y = 2x + 5 gives x = (y - 5)/2, which belongs to R.
  4. 4Hence f is bijective and invertible.
  5. 5To find inverse, y = 2x + 5, so x = (y - 5)/2.
  6. 6Therefore f⁻¹(y) = (y - 5)/2, and f⁻¹(x) = (x - 5)/2.

Method you can reuse

  1. 1Note domain and codomain.
  2. 2Prove one-one using f(a) = f(b), or give a counterexample if disproving.
  3. 3Prove onto by solving y = f(x) for x in the domain.
  4. 4Once bijective, solve y = f(x) for x.
  5. 5Write the inverse with correct domain and codomain.
  6. 6Optionally verify using f⁻¹(f(x)) = x and f(f⁻¹(x)) = x.

Visual tip

A reverse mapping diagram may support the idea of inverse, but the exam-useful test is bijectivity plus composition identity.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is f: R to R, f(x) = x2 not invertible?

Think first, then show answer
It is not one-one because f(1) = f(-1) = 1 for different inputs. Therefore it is not bijective and not invertible from R to R.

Common wrong answer

Forgetting that f⁻¹ has domain equal to the codomain of f, causing an inverse formula to be written with the wrong domain.

Don't mix this up

Students often find an algebraic reverse formula and call it an inverse without proving or checking that the original function is bijective on the stated domain and codomain.

Exam tip

Before writing f⁻¹, state why f is one-one and onto, or use the given fact that it is bijective. For restricted domains, keep the restricted range in mind.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as proof of invertibility, finding the inverse of a given function, or verifying that two functions are inverses using composition. Assertion-reason questions often test the statement that every bijection has an inverse.
  1. 1

    Enough for 1 mark

    A function f: A to B is invertible if there exists a function f⁻¹: B to A such that f⁻¹(f(x)) = x for every x in A and f(f⁻¹(y)) = y for every y in B. A function is invertible if and only if it is bijective.

  2. 2

    Add for 2 marks

    Example: For f: R to R, f(x) = 5x - 7, set y = 5x - 7. Then x = (y + 7)/5. Hence f⁻¹(y) = (y + 7)/5, or f⁻¹(x) = (x + 7)/5.

  3. 3

    Add for 3 marks

    An inverse function reverses the action of the original function.

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