GCSE Maths Algebra - How to Solve: Function Notation and Inverse Functions

How to Solve: Function Notation and Inverse Functions

GCSE / A-Level (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering function notation and inverses lets you decode exam questions worth 5–10 marks—like predicting drug concentration in biology or calculating resistor values in physics. One wrong step, and you lose all the marks. Today, you’ll learn the exact method to solve them every time."

WHAT YOU NEED TO KNOW FIRST

1. Basic algebra – Rearranging equations (e.g., solving for x).
2. Substitution – Plugging numbers into expressions (e.g., f(2) means replace x with 2).
3. Graphs of functions – Recognising linear (y = mx + c) and quadratic (y = ax² + bx + c) shapes.

KEY TERMS & FORMULAS

1. Function Notation

• Term: f(x) = "f of x" = a rule that takes an input x and gives an output.
• Example: If f(x) = 3x + 2, then f(4) = 3(4) + 2 = 14.
• MEMORISE THIS: f(a) means "substitute x = a into the function."

2. Inverse Functions

• Term: f⁻¹(x) = "f inverse of x" = reverses the original function f(x).
• Formula: If y = f(x), then x = f⁻¹(y).
• MEMORISE THIS: To find f⁻¹(x):
• Write y = f(x).
• Swap x and y.
• Rearrange to make y the subject.
• Replace y with f⁻¹(x).

3. Domain & Range

• Domain: All possible x values (inputs) for the function.
• Range: All possible y values (outputs) from the function.
• Given on exam sheet: Check if the question restricts domain (e.g., x ≥ 0).

STEP-BY-STEP METHOD

Part 1: Evaluating Functions (f(x))

Step 1: Identify the function rule (e.g., f(x) = 2x – 5). Step 2: Replace x with the given input (e.g., f(3) → 2(3) – 5). Step 3: Calculate the result (e.g., 6 – 5 = 1).

Part 2: Finding Inverse Functions (f⁻¹(x))

Step 1: Write y = f(x) (e.g., y = 4x + 1). Step 2: Swap x and y (e.g., x = 4y + 1). Step 3: Rearrange to solve for y (e.g., x – 1 = 4y → y = (x – 1)/4). Step 4: Write f⁻¹(x) (e.g., f⁻¹(x) = (x – 1)/4).

Part 3: Checking Inverses

Step 1: Compose f(f⁻¹(x)) or f⁻¹(f(x)). Step 2: Simplify. If the result is x, the inverse is correct.

WORKED EXAMPLES

Example 1 – Basic: Evaluating f(x)

Question: If f(x) = 5x – 7, find f(2). Solution: 1. f(2) = 5(2) – 7 2. f(2) = 10 – 7 3. f(2) = 3 What we did and why: Substituted x = 2 into the function and simplified.

Example 2 – Medium: Finding f⁻¹(x)

Question: Find the inverse of f(x) = 3x + 6. Solution: 1. y = 3x + 6 2. Swap x and y: x = 3y + 6 3. Rearrange: x – 6 = 3y → y = (x – 6)/3 4. f⁻¹(x) = (x – 6)/3 What we did and why: Followed the inverse steps to reverse the function.

Example 3 – Exam-Style: Combined Function & Inverse

Question: Given f(x) = 2x – 4 and g(x) = x², find f⁻¹(g(3)). Solution: 1. Find g(3): g(3) = 3² = 9 2. Find f⁻¹(x):
- y = 2x – 4
- Swap: x = 2y – 4
- Rearrange: x + 4 = 2y → y = (x + 4)/2
- f⁻¹(x) = (x + 4)/2 3. Substitute g(3) = 9 into f⁻¹: f⁻¹(9) = (9 + 4)/2 = 13/2 = 6.5 What we did and why: Broke the problem into smaller steps—evaluated g(3), found f⁻¹, then composed them.

COMMON MISTAKES

1. MISTAKE: Forgetting to swap x and y when finding inverses.
   WHY IT HAPPENS: Confusing f(x) with f⁻¹(x).
   CORRECT APPROACH: Always swap x and y first.

2. MISTAKE: Incorrectly substituting values (e.g., f(2x) instead of f(x)).
   WHY IT HAPPENS: Misreading the input.
   CORRECT APPROACH: Circle the input (e.g., f(2) → replace x with 2).

3. MISTAKE: Not simplifying fully (e.g., leaving f⁻¹(x) = (x + 3)/2 as x/2 + 3/2).
   WHY IT HAPPENS: Rushing the algebra.
   CORRECT APPROACH: Always simplify fractions.

4. MISTAKE: Assuming all functions have inverses (e.g., f(x) = x² fails the horizontal line test).
   WHY IT HAPPENS: Ignoring domain restrictions.
   CORRECT APPROACH: Check if the function is one-to-one (passes horizontal line test).

5. MISTAKE: Mixing up f⁻¹(x) with 1/f(x).
   WHY IT HAPPENS: Misinterpreting the notation.
   CORRECT APPROACH: f⁻¹(x) reverses the function, 1/f(x) is the reciprocal.

EXAM TRAPS

1. TRAP: Questions ask for f⁻¹(5) but give f(x) in a word problem.
   HOW TO SPOT IT: Look for phrases like "reverse the process" or "find the original input."
   HOW TO AVOID IT: Write y = f(x), swap, and solve for y when you see f⁻¹.

2. TRAP: Functions with restricted domains (e.g., f(x) = √x only defined for x ≥ 0).
   HOW TO SPOT IT: The question mentions "for x ≥ 0" or shows a graph.
   HOW TO AVOID IT: Always check the domain before finding inverses.

3. TRAP: Composite functions (e.g., f(g(x))) disguised as a single step.
   HOW TO SPOT IT: The question says "apply f then g" or vice versa.
   HOW TO AVOID IT: Break it into two steps: evaluate the inner function first.

1-MINUTE RECAP

"Here’s the night-before cheat sheet: 1. Evaluating f(x): Substitute the number into x and simplify. 2. Finding f⁻¹(x): Write y = f(x), swap x and y, rearrange for y, then rename as f⁻¹(x). 3. Checking inverses: Compose f(f⁻¹(x))—if you get x, it’s correct. 4. Watch for traps: Swap x and y every time, simplify fully, and check domains. 5. Exam questions: Break them into steps—evaluate first, then find inverses, then compose.

You’ve got this. Now go ace that exam!"