Week 12: Functions - Revision Notes
1. Function Notation
What is a function?
A function is a rule that takes an input (x) and produces an output.
f(x) means: the output when the input is x
Example 1: Evaluating a function
If f(x) = 2x + 3, find f(5)
Step 1: Replace x with 5
f(5) = 2(5) + 3
Step 2: Calculate
f(5) = 10 + 3 = 13
Answer: f(5) = 13
Example 2: Solving f(x) = k
If f(x) = 3x - 4, solve f(x) = 11
Step 1: Write the equation
3x - 4 = 11
Step 2: Solve for x
3x = 15
x=5
Answer: x = 5
2. Domain and Range
Domain: all possible input values (x-values)
Range: all possible output values (y-values)
Example 3: Finding range
For f(x) = x² with domain 0 ≤ x ≤ 3, find the range
When x = 0: f(0) = 0² = 0
When x = 3: f(3) = 3² = 9
Range: 0 ≤ f(x) ≤ 9
3. Inverse Functions
The inverse function reverses what the original function does.
Method to find inverse:
1. Write y = f(x)
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2. Swap x and y
3. Solve for y
4. Write answer as f⁻¹(x)
Example 4: Finding inverse
If f(x) = 2x + 3, find f⁻¹(x)
Step 1: y = 2x + 3
Step 2: Swap - x = 2y + 3
Step 3: Solve for y
x - 3 = 2y
y = (x - 3)/2
Answer: f⁻¹(x) = (x - 3)/2
4. Composite Functions
fg(x) means: do g first, then do f to the result
fg(x) = f(g(x))
Example 5: Finding composite function
If f(x) = 2x and g(x) = x + 3, find fg(x)
Step 1: Start with g(x)
g(x) = x + 3
Step 2: Apply f to g(x)
f(g(x)) = f(x + 3) = 2(x + 3)
Step 3: Simplify
fg(x) = 2x + 6
WARNING: fg(x) ≠ gf(x)
Order matters! Always work from inside out.
Common Exam Mistakes
Mistake 1: Thinking fg(x) = f(x) × g(x)
fg(x) is NOT multiplication - it's substitution!
Mistake 2: Not swapping x and y when finding inverse
You must swap them or your answer will be wrong
Mistake 3: Confusing fg(x) with gf(x)
These are different! Order matters.
Practice Questions
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1. If f(x) = 3x - 5, find f(4)
2. If f(x) = x² + 2, solve f(x) = 11
3. If f(x) = 4x + 1, find f⁻¹(x)
4. If f(x) = 3x and g(x) = x - 2, find fg(x)
5. Using functions from Q4, find gf(x)
Answers:
1. f(4) = 7
2. x = 3 or x = -3
3. f⁻¹(x) = (x - 1)/4
4. fg(x) = 3x - 6
5. gf(x) = 3x - 2
Your Homework
Complete by next Monday:
✓ Rayner 8.6 all exercises
✓ Past paper questions on Google Classroom
✓ Post questions if stuck - I'll help within 24 hours
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