Functions (Copy)
FUNCTIONS, COMPOSITE FUNCTIONS, MODULUS FUNCTIONS, GRAPHS & INVERSE FUNCTIONS — FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Function Notation | f(x) = output for input x | Treating f(x) as multiplication |
| Composite Function | (f ∘ g)(x) = f(g(x)) | Doing functions in wrong order |
| Composite Function Reverse | (g ∘ f)(x) = g(f(x)) | Assuming both composites are equal |
| Inverse Function Rule | f(f⁻¹(x)) = x | Forgetting verification |
| Reflection Property | f(x) and f⁻¹(x) reflect in y = x | Reflecting in x-axis |
| Coordinate Reflection | (a, b) → (b, a) | Not swapping coordinates |
| Horizontal Line Test | Used to check inverses | Using vertical line test instead |
| Vertical Line Test | Used to identify functions | Using it for inverse checking |
| Linear Function | y = mx + c | Mixing m and c |
| Gradient Formula | m = (y₂ − y₁)/(x₂ − x₁) | Reversing order incorrectly |
| Quadratic Function | y = ax² + bx + c | Forgetting negative signs |
| Axis of Symmetry | x = −b/2a | Missing brackets |
| Discriminant | b² − 4ac | Forgetting square on b |
| Nature of Roots | b² − 4ac > 0 → 2 roots | Mixing conditions |
| Vertex Form | y = a(x − h)² + k | Wrong sign inside bracket |
| Modulus Definition | |x| = x if x ≥ 0 | Ignoring piecewise definition |
| Modulus Definition | |x| = −x if x < 0 | Forgetting sign reversal |
| Modulus Equation | |x| = a → x = ±a | Giving one answer only |
| Modulus Graph | V-shaped graph | Drawing parabola |
| Even Function | f(−x) = f(x) | Mixing with odd function |
| Odd Function | f(−x) = −f(x) | Forgetting negative sign |
| Domain Restriction | Denominator ≠ 0 | Allowing undefined values |
| Root Restriction | Inside square root ≥ 0 | Using negatives inside roots |
| Inverse Step | Swap x and y | Forgetting swap |
| Inverse Step | Solve fully for y | Leaving mixed variables |
| Domain & Range | Domain of f = Range of f⁻¹ | Forgetting interchange |
| Quadratic Inverse | Restrict domain first | Inverting full parabola |
| Exponential Function | y = aˣ | Writing xᵃ accidentally |
| Logarithmic Function | y = logₐ(x) | Forgetting x > 0 |
| Log Rule | logₐ(mn) = logₐm + logₐn | Adding inside logs |
| Log Rule | logₐ(m/n) = logₐm − logₐn | Dividing logs incorrectly |
| Log Rule | logₐ(mⁿ) = nlogₐm | Forgetting power rule |
| Reflection in x-axis | y = −f(x) | Reflecting in y-axis |
| Reflection in y-axis | y = f(−x) | Reflecting in x-axis |
| Vertical Stretch | y = af(x) | Confusing with horizontal |
| Horizontal Stretch | y = f(bx) | Wrong direction |
| Trig Restriction | sin(x): −π/2 ≤ x ≤ π/2 | Using unrestricted domain |
| Reciprocal Function | y = 1/x | Forgetting asymptotes |
| Exponential Inverse | aˣ ↔ logₐ(x) | Mixing inverse pairs |
| Intersections | Solve f(x) = g(x) | Equating wrong functions |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK GRAPH FEATURES
| Graph | Shape | Important Feature |
|---|---|---|
| y = x | Straight line | Gradient = 1 |
| y = x² | Parabola | Vertex at (0,0) |
| y = x³ | Cubic | S-shaped |
| y = |x| | V-shape | Vertex at origin |
| y = √x | Half curve | Starts at origin |
| y = 1/x | Hyperbola | Two asymptotes |
| y = aˣ | Exponential | Horizontal asymptote |
| y = logₐ(x) | Log curve | Vertical asymptote |
QUICK INVERSE FUNCTION STEPS
| Step | Action |
|---|---|
| 1 | Write y = f(x) |
| 2 | Swap x and y |
| 3 | Solve for y |
| 4 | Replace y with f⁻¹(x) |
| 5 | Verify using composites |
QUICK MODULUS SOLVING STEPS
| Step | Action |
|---|---|
| 1 | Remove modulus using cases |
| 2 | Solve each equation |
| 3 | Reject invalid solutions |
| 4 | Write final answers |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK COMPOSITE FUNCTION STEPS
| Step | Action |
|---|---|
| 1 | Solve inner function first |
| 2 | Substitute carefully |
| 3 | Expand brackets correctly |
| 4 | Simplify fully |
GRAPH TRANSFORMATION RULES
| Transformation | Effect |
|---|---|
| y = f(x) + k | Up by k |
| y = f(x) − k | Down by k |
| y = f(x + a) | Left by a |
| y = f(x − a) | Right by a |
| y = −f(x) | Reflect in x-axis |
| y = f(−x) | Reflect in y-axis |
| y = af(x) | Vertical stretch/compression |
| y = f(bx) | Horizontal stretch/compression |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Inverse graph | Reflect in y = x |
| Modulus graph | Reflect negative part upward |
| Quadratic turning point | Use x = −b/2a |
| Composite functions | Inside first, outside second |
| Transformation signs | Inside opposite, outside normal |
| Even/Odd test | Substitute −x immediately |
| Domain questions | Check denominator + roots first |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON EXAM LOSSES OF MARKS
| Mistake | Why Students Lose Marks |
|---|---|
| Forgetting restricted domain | Wrong inverse |
| Missing ± in modulus | Incomplete answers |
| Wrong transformation direction | Incorrect graph |
| Using vertical instead of horizontal line test | Wrong conclusion |
| Forgetting asymptotes | Incomplete graph |
| Wrong order in composite functions | Entire answer wrong |
| Forgetting brackets in substitution | Algebra errors |
| Mixing odd/even definitions | Incorrect symmetry |
| Ignoring domain restrictions in logs | Undefined answers |
| Leaving inverse unsimplified | Method marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.