Functions and Graph Transformations
Functions
A function maps each input to exactly one output. Key terms:
- Domain = the set of allowed inputs (x-values); range = the set of possible outputs (y-values).
- A function must be one-to-one (or restricted to be) to have an inverse.
Composite functions
fg(x) means "do g first, then f": fg(x) = f(g(x)). Order matters — fg is generally not the same as gf.
- Example: f(x) = 2x, g(x) = x + 3 → fg(x) = 2(x + 3) = 2x + 6; gf(x) = 2x + 3.
Inverse functions
The inverse f⁻¹(x) reverses f: if f(a) = b then f⁻¹(b) = a.
- Method: write y = f(x), swap x and y, rearrange to make y the subject.
- The graph of f⁻¹ is the reflection of f in the line y = x.
- The domain of f⁻¹ = the range of f (and vice versa).
The modulus function
|x| gives the positive value (distance from zero). To solve |f(x)| = a, solve f(x) = a and f(x) = −a. The graph of y = |f(x)| reflects any part below the x-axis above it.
Graph transformations
For y = f(x):
| Transformation | Effect |
|---|---|
| f(x) + a | translate up by a |
| f(x + a) | translate left by a (opposite sign) |
| af(x) | vertical stretch, scale factor a |
| f(ax) | horizontal stretch, scale factor 1/a |
| −f(x) | reflect in the x-axis |
| f(−x) | reflect in the y-axis |
Inside the bracket → horizontal (and "backwards"); outside → vertical (as expected).
Worked example
Given f(x) = 3x − 1, find f⁻¹(x).
- y = 3x − 1 → swap: x = 3y − 1 → x + 1 = 3y → y = (x + 1)/3.
- f⁻¹(x) = (x + 1)/3. ✓
Common mistakes
- Doing fg as "f then g" — it's g first, then f.
- Getting horizontal translations the wrong way: f(x + a) moves left.
- Forgetting a function must be one-to-one for an inverse to exist.
Exam tips
- State domain and range, especially for inverses (they swap).
- Apply transformations in the right order; inside the bracket affects x (horizontally, reversed).
- For modulus equations, consider both the positive and negative cases.
Key facts to remember
- Composite fg(x) = f(g(x)) (g first); inverse found by swapping x and y — reflection in y = x; domain/range swap.
- |x| = positive value; solve |f(x)| = a via f(x) = ±a.
- Transformations: outside bracket = vertical (as written); inside bracket = horizontal and reversed.