Functions, Graphs, and Transformations
Core concepts
A function assigns each input exactly one output. You must be fluent with domain and range, function notation, composite functions, and inverses. For inverses, remember that the domain of f becomes the range of f⁻¹ and vice versa; a function has an inverse function only if it is one-to-one on its stated domain.
Transformations are best understood from the parent graph y = f(x). The form y = a·f(b(x − h)) + k tells you: a scales vertically (reflection in the x-axis if a < 0), b scales horizontally (reflection in the y-axis if b < 0), h shifts horizontally, and k shifts vertically. Horizontal changes are the counterintuitive ones: y = f(x − 3) shifts right by 3, and y = f(2x) compresses horizontally by a factor of ½, not 2.
Traps to watch
- Applying transformations in the wrong order when several are combined; factor out b before reading h.
- Confusing f(x) + k (vertical shift) with f(x + k) (horizontal shift).
- Forgetting to restrict the domain when forming an inverse of a function that is not one-to-one, such as a quadratic.
- Stating the domain without excluding values that make a denominator zero or a square root negative.
Rule of thumb
Inside the brackets means horizontal and behaves opposite to intuition; outside means vertical and behaves as expected. Sketch the parent graph first, then move key points one transformation at a time.
Worked example
For y = −2√(x + 1) + 4, start with y = √x, shift left 1, stretch vertically by 2, reflect in the x-axis, then shift up 4. The endpoint (0, 0) maps to (−1, 4), and the graph opens downward from there.