Functions (Copy)
1. Key Terms & Definitions
| Term | Meaning | Example |
|---|---|---|
| Function | A rule mapping each input (x) to exactly one output (y). | f(x) = x² + 1 |
| Domain | Set of all possible input values. | For f(x) = √x, Domain: x ≥ 0 |
| Range / Image set | Set of all possible output values. | For f(x) = x², Range: y ≥ 0 |
| One–one function | Each x maps to a unique y, and each y comes from exactly one x. | f(x) = 2x + 3 |
| Many–one function | Different x-values can give the same y-value. | f(x) = x² |
| Inverse function (f⁻¹) | Reverses the mapping of f. Exists only for one–one functions. | f(x) = 3x − 2 → f⁻¹(x) = (x + 2)/3 |
| Composite function (fg) | Apply g first, then f. | If f(x) = 2x, g(x) = x + 3 → fg(x) = f(g(x)) = 2(x + 3) |
2. Function Notation
- f(x) means “value of f when input is x”.
- Inverse: f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
- Composite: fg(x) = f(g(x)), gf(x) = g(f(x)) — order matters.
- Repeated function: f²(x) = f(f(x)) (not squaring!).
3. Domain & Range Rules
| Step | Action |
|---|---|
| 1 | Start with original function. |
| 2 | Identify restrictions (e.g., no division by 0, no square root of negative). |
| 3 | Domain of composite gf ⊆ Domain of f. |
| 4 | Range of gf ⊆ Range of g. |
| 5 | For inverse: restrict domain so f is one–one. |
Example: f(x) = √(x − 3)
- Domain: x ≥ 3
- Range: y ≥ 0
4. Relationship Between y = f(x) and y = |f(x)|
| Case | Effect on Graph |
|---|---|
| f(x) ≥ 0 | Graph unchanged. |
| f(x) < 0 | Reflect negative part above x-axis. |
Example: y = |x − 2|
- Same as y = x − 2 for x ≥ 2
- Reflected for x < 2.
5. Inverse Functions
Steps to find f⁻¹(x):
- Replace f(x) with y.
- Swap x and y.
- Solve for y.
- Rename y as f⁻¹(x).
Example: f(x) = e²ˣ
- y = e²ˣ
- ln y = 2x → x = (1/2) ln y
- f⁻¹(x) = (1/2) ln x
6. Composite Functions Example
f(x) = 2x + 1, g(x) = x²
- fg(x) = f(g(x)) = 2x² + 1
- gf(x) = g(f(x)) = (2x + 1)² = 4x² + 4x + 1
Note: fg ≠ gf.
7. Graphs of f and f⁻¹
- f and f⁻¹ are mirror images in the line y = x.
- Domain of f → Range of f⁻¹
- Range of f → Domain of f⁻¹
8. Common Mistakes to Avoid
| Mistake | Correct Approach |
|---|---|
| Forgetting to restrict domain for inverse. | Check one–one condition first. |
| Mixing fg and gf. | Apply inside function first. |
| Thinking f²(x) means squaring. | Remember: f²(x) = f(f(x)). |
| Ignoring | f(x) |
| Swapping x & y but forgetting to solve fully for y in inverse. | Always rearrange to isolate y. |
9. Quick Example Table
| Function | Domain | Range | One–one? | Inverse? |
|---|---|---|---|---|
| f(x) = x² | ℝ | y ≥ 0 | No | Not without domain restriction |
| f(x) = 2x + 5 | ℝ | ℝ | Yes | Yes |
| f(x) = √(x − 1) | x ≥ 1 | y ≥ 0 | Yes | Yes |
A. Graph Transformations for Functions
| Transformation | Rule | Graph Effect |
|---|---|---|
| y = f(x) + k | Add k outside | Shift up k units (k > 0) or down k units (k < 0) |
| y = f(x − h) | Subtract h inside | Shift right h units (h > 0) or left h units (h < 0) |
| y = −f(x) | Multiply output by −1 | Reflect in x-axis |
| y = f(−x) | Replace x with −x | Reflect in y-axis |
| **y = | f(x) | ** |
| y = af(x) | Multiply output by a | Vertical stretch ( |
| y = f(bx) | Multiply input by b | Horizontal compression ( |
| Inverse y = f⁻¹(x) | Swap x & y | Reflection in line y = x |
Example: f(x) = x²
- y = |f(x)| → Same as x² (already non-negative)
- y = f⁻¹(x) (domain x ≥ 0) → y = √x, mirror image in y = x
B. Kinematics Graph Relationships
| Starting Graph | How to Get Next Graph | Graph Shape Change |
|---|---|---|
| s–t (displacement–time) | Slope = velocity (v–t) | Positive slope = moving forward, negative slope = moving backward |
| v–t (velocity–time) | Slope = acceleration (a–t) | Horizontal line = constant velocity (a = 0) |
| a–t (acceleration–time) | Area = change in velocity | Area under curve adds to velocity |
| v–t (velocity–time) | Area = displacement change | Area above x-axis = forward displacement, below = backward displacement |
Kinematics Example
s(t) = 3t³ − 10t² + 4t + 8 (0 ≤ t ≤ 3)
- Velocity: v(t) = 9t² − 20t + 4 → Parabola opening upward
- Acceleration: a(t) = 18t − 20 → Straight line
- Graph flow: s–t (cubic) → v–t (quadratic) → a–t (linear)
C. Visual Summary
Functions:
f(x) → |f(x)| = reflect negative y-values above x-axis
f(x) → f⁻¹(x) = reflect across y = x
Kinematics:
s–t → slope → v–t → slope → a–t
Integrating reverses: a–t → area → v–t → area → s–t