How to Solve: Function Notation and Inverse Functions

How to Solve: Function Notation and Inverse Functions

GCSE / A-Level Maths

Introduction

"Mastering function notation and inverses lets you decode real-world relationships—like reversing a temperature conversion or cracking an encryption code—and it’s worth 10-15% of your exam marks in algebra questions. Miss this, and you’ll lose easy points on graphs, equations, and even calculus later."

What You Need To Know First

1. Substitution: Plugging numbers into expressions (e.g., if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11).
2. Rearranging equations: Solving for y in terms of x (e.g., y = 3x – 5 → x = (y + 5)/3).
3. Graphs of linear functions: Straight lines, y = mx + c, and how to reflect them.

Key Vocabulary

Formulas To Know

1. Function notation
2. f(x) = [expression]

3. MEMORISE THIS: f(a) means "substitute x = a into the function."

4. Inverse function (linear)

5. If f(x) = ax + b, then f⁻¹(x) = (x – b)/a.

6. MEMORISE THIS: Swap x and y, then solve for y.

7. Inverse function (non-linear)

8. Given y = f(x), swap x and y, then rearrange to y = f⁻¹(x).

9. MEMORISE THIS: Always check if the function is one-to-one first!

10. Composite functions

11. (f ∘ g)(x) = f(g(x))
12. MEMORISE THIS: Work from the inside out.

Step-by-Step Method

Part 1: Evaluating Functions (f(x) = ...)

1. Write down the function (e.g., f(x) = 3x – 7).
2. Replace every x with the given input (e.g., f(5) = 3(5) – 7).
3. Simplify (e.g., 15 – 7 = 8).
4. Write the final answer (e.g., f(5) = 8).

Part 2: Finding Inverse Functions

1. Write the function as y = ... (e.g., y = 4x + 1).
2. Swap x and y (e.g., x = 4y + 1).
3. Solve for y:
4. Subtract 1: x – 1 = 4y.
5. Divide by 4: y = (x – 1)/4.
6. Write the inverse in function notation (e.g., f⁻¹(x) = (x – 1)/4).
7. Check it’s one-to-one (if not, restrict the domain).

Part 3: Using Inverses to Solve Equations

1. Write the equation (e.g., f(x) = 10, where f(x) = 2x + 3).
2. Apply the inverse function to both sides (e.g., f⁻¹(f(x)) = f⁻¹(10)).
3. Simplify (e.g., x = f⁻¹(10)).
4. Calculate the inverse (e.g., f⁻¹(x) = (x – 3)/2, so x = (10 – 3)/2 = 3.5).

Worked Examples

Example 1 – Basic: Evaluating a Function

Question: If f(x) = 5x – 2, find f(3). Steps: 1. Write the function: f(x) = 5x – 2. 2. Substitute x = 3: f(3) = 5(3) – 2. 3. Simplify: 15 – 2 = 13. Answer: f(3) = 13. What we did and why: We replaced x with 3 and simplified to find the output. This is the foundation of function notation.

Example 2 – Medium: Finding an Inverse Function

Question: Find the inverse of f(x) = 7x – 4. Steps: 1. Write as y = 7x – 4. 2. Swap x and y: x = 7y – 4. 3. Solve for y:
- Add 4: x + 4 = 7y.
- Divide by 7: y = (x + 4)/7. 4. Write in function notation: f⁻¹(x) = (x + 4)/7. Answer: f⁻¹(x) = (x + 4)/7. What we did and why: We swapped x and y to "undo" the function, then rearranged to find the inverse.

Example 3 – Exam-Style: Composite Functions & Inverses

Question: Given f(x) = 2x + 1 and g(x) = x², solve f(g(x)) = 9. Steps: 1. Find f(g(x)):
- g(x) = x², so f(g(x)) = f(x²) = 2x² + 1. 2. Set equal to 9: 2x² + 1 = 9. 3. Solve for x:
- Subtract 1: 2x² = 8.
- Divide by 2: x² = 4.
- Square root: x = ±2. Answer: x = 2 or x = –2. What we did and why: We composed the functions first, then solved the equation. This tests both function notation and inverse thinking.

Common Mistakes

1. Mistake: Forgetting to swap x and y when finding inverses.
   WHY IT HAPPENS: Students rush and treat it like a normal equation.
   CORRECT APPROACH: Always swap x and y first—this is the "undoing" step.

2. Mistake: Not simplifying fully (e.g., leaving f⁻¹(x) = (x – 3)/2 as y = x/2 – 3/2).
   WHY IT HAPPENS: Students stop halfway through rearranging.
   CORRECT APPROACH: Keep simplifying until y is isolated.

3. Mistake: Assuming all functions have inverses (e.g., f(x) = x²).
   WHY IT HAPPENS: Forgetting that inverses only work for one-to-one functions.
   CORRECT APPROACH: Check if the function passes the horizontal line test (or restrict the domain).

4. Mistake: Misapplying function notation (e.g., writing f(x + 2) = f(x) + 2).
   WHY IT HAPPENS: Confusing f with multiplication.
   CORRECT APPROACH: Substitute the whole expression: f(x + 2) = 3(x + 2) – 1 (if f(x) = 3x – 1).

5. Mistake: Forgetting to check answers (e.g., plugging x = 2 back into f(g(x))).
   WHY IT HAPPENS: Overconfidence in algebra.
   CORRECT APPROACH: Always verify by substituting your answer back into the original equation.

Exam Traps

1. Trap: Giving a function that’s not one-to-one (e.g., f(x) = x²) and asking for an inverse without a restricted domain.
   How to Spot it: The question mentions "inverse" but the function fails the horizontal line test.
   How to Avoid it: State the domain restriction (e.g., x ≥ 0) or explain why the inverse doesn’t exist.

2. Trap: Using f⁻¹(x) to mean 1/f(x) (e.g., thinking f⁻¹(x) = 1/(2x + 3)).
   How to Spot it: The question uses f⁻¹ but the answer looks like a reciprocal.
   How to Avoid it: Remember: f⁻¹ means "inverse function," not "reciprocal."

3. Trap: Composite functions with hidden steps (e.g., f(g(x)) where g(x) is a fraction or root).
   How to Spot it: The question has nested functions (e.g., f(√x)).
   How to Avoid it: Work from the inside out, substituting carefully.

1-Minute Recap

"Listen up—this is your 60-second cheat sheet for function notation and inverses. First, f(x) is just a rule: plug in x, get an output. To find f(5), replace every x with 5 and simplify. For inverses, swap x and y, then solve for y—that’s your f⁻¹(x). Always check if the function is one-to-one, or you’ll lose marks. In exams, watch out for tricks like non-one-to-one functions or composite functions. If you see f(g(x)), work from the inside out. And remember: f⁻¹ is the inverse, not the reciprocal. Now go practice—you’ve got this!