Grasp Maths

Year 11

Functions, iteration and proof-style reasoning

Use formal function notation accurately, explain recurring steps in iteration and sequences, and recognise the structure of proof-style questions.

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Lesson overview

Algebra — functions and sequences

A function f(x) maps an input x to an output f(x). Composite functions combine functions: f(g(x)) means apply g first, then f. Inverse functions f−1{f}^{-1}(x) reverse the mapping. Iteration is repeated application of a function: f(f(x)) or f(f(f(x)))... In proof questions, you are often asked to show that a statement is always true for a class of numbers. Use algebra to justify general claims, and always explain why each step is valid. Key structures include: showing an expression is always even/odd, always divisible by a number, or always satisfies a given relationship.

Evaluating composite functions

If f(x) = 2x + 1 and g(x) = x2{x}^{2}, find f(g(3)).

Apply g first: 32{3}^{2} = 9. Then apply f: f(9) = 2(9) + 1 = 19.

Finding an inverse function

If f(x) = 3x − 5, find f−1{f}^{-1}(x).

Write y = f(x), rearrange to make x the subject, swap x and y.

Iteration — applying a function repeatedly

If f(x) = 2x − 1, find f(f(2)).

First iteration: f(2) = 3. Second iteration: f(3) = 5.

Proof: show an expression is always divisible by a number

Show that n(n + 1) is always even, where n is any integer.

Use the property that one of two consecutive integers is always even.

Worked example — composite functions and proof

If f(x) = x + 2 and g(x) = 3x, write down f(g(x)) and g(f(x)). Show that f(g(x)) ≠ g(f(x)) in general.

  1. Apply g to x: g(x) = 3x.
  2. Apply f to the result: f(g(x)) = f(3x) = 3x + 2.
  3. Apply f to x: f(x) = x + 2.
  4. Apply g to the result: g(f(x)) = g(x + 2) = 3(x + 2) = 3x + 6.
  5. Compare: f(g(x)) = 3x + 2 and g(f(x)) = 3x + 6.
  6. Conclusion: f(g(x)) − g(f(x)) = −4 ≠ 0, so composition is not commutative.

Try it

Use the examples carefully, then choose the answer.

Question 1

If f(x) = 2x and g(x) = x − 3, what is f(g(5))?

💡 g(5) = 5 − 3 = 2. Then f(2) = 2(2) = 4.

Question 2

If f(x) = x2{x}^{2} + 1, what is f(f(1))?

💡 f(1) = 12{1}^{2} + 1 = 2. Then f(2) = 22{2}^{2} + 1 = 5.

Question 3

If f(x) = 4x − 1, what is f−1{f}^{-1}(x)?

💡 Reverse: y = 4x − 1 → y + 1 = 4x → x = (y + 1)/4.

Question 4

Show that (2n − 1)(2n + 1) = 4n2{n}^{2} − 1. Which step justifies using difference of squares?

💡 Recognise a2{a}^{2} − b2{b}^{2} = (a − b)(a + b) pattern with a = 2n, b = 1.

Question 5

Prove that n2{n}^{2} − n is always even for any integer n. Which fact do you use?

💡 n2{n}^{2} − n = n(n − 1). One of two consecutive integers is always even.

Common mistakes

Watch for these when working through the lesson.

  • Mixing up the order in composite functions: f(g(x)) means apply g first, not f first.
  • Forgetting to fully rearrange when finding inverse functions; the answer should be in terms of x.
  • In proofs, using specific examples instead of general algebra ('Let n be any integer...' then argue generally).
  • Not justifying steps in a proof; each transformation must be explained or justified by a property.

Related topics

These ideas fit closely with this lesson.

  • Algebraic manipulation and rearrangement
  • Solving quadratic and linear equations
  • Sequences and series
  • Showing expressions are always equal

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.