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Paper 2
Functions & Graphs
Both
Use graphs to sketch function behaviour quickly.
Key Facts
- Composite: f(g(x)) means do g then f.
- Inverse undoes the function.
- Transformations shift or stretch graphs.
- Domain and range can be restricted.
Key Equations
f(f^-1(x)) = x
f^-1(f(x)) = x
Topics Covered
Function Notation
What you need to know
- •f(x) represents an output based on x.
- •Composite functions combine two functions.
- •f(g(x)) means apply g first, then apply f to the result.
- •Domain is the set of inputs, range is the set of outputs.
Exam Tips
- Write intermediate step when evaluating composites.
- Check domain restrictions.
Inverse Functions
What you need to know
- •f^-1(x) undoes what f(x) does.
- •To find inverse: replace f(x) with y, swap x and y, solve for y.
- •f(f^-1(x)) = x and f^-1(f(x)) = x.
Exam Tips
- Check your inverse by composing with the original.
Function Transformations
What you need to know
- •f(x) + a shifts graph up by a.
- •f(x + a) shifts graph left by a.
- •af(x) stretches vertically by factor a.
- •f(ax) stretches horizontally by factor 1/a.
Exam Tips
- Sketch transformations step by step.
Key Terms
Composite function
A function made by applying one function to another.
Inverse function
Function that undoes another: f^-1(f(x)) = x.
Domain
Set of all possible input values.
Range
Set of all possible output values.
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Common Exam Questions
Given f(x)=2x-1 and g(x)=x^2, find f(g(3)).
2 markseasyPaper 2
Model Answer
g(3) = 9, f(9) = 18 - 1 = 17
What examiners want to see
- ✓Apply g then f in order.
- ✓Show working.
Find the inverse of f(x) = 3x + 5.
3 marksmediumPaper 2
Model Answer
y = 3x + 5. Swap: x = 3y + 5. Solve: y = (x - 5)/3. So f^-1(x) = (x - 5)/3.
What examiners want to see
- ✓Swap x and y.
- ✓Solve for y.
Describe the transformation from y = x^2 to y = 2x^2 + 3.
2 marksmediumPaper 2
Model Answer
Vertical stretch by factor 2, then shift up 3 units.
What examiners want to see
- ✓Identify both transformations.
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