Functions: Domain, Range, and Transformations Review
Review function notation, domain, range, and the library of parent functions. Revisit transformations (shifts, reflections, stretches) as a framework that applies to every function class in the course.
A lesson of Algebra 2, in Review: Linear, Quadratic & Complex Numbers, about 10 minutes.
By the end of this lesson, you will be able to identify domain and range of functions from equations and graphs, apply the six transformation types (shifts, reflections, stretches), compose functions, and determine if a function is invertible.
The Big Idea
Algebra 2 builds on one foundational concept: the function. Before introducing new function families (polynomials, rationals, exponentials), we sharpen our tools for analyzing any function: its input-output behavior (domain/range), its visual transformations, and how functions combine with each other.
Domain and Range
Domain: the set of all valid inputs (x-values).
Range: the set of all outputs (y-values) the function produces.
Restricting the domain:
- Division by zero: exclude values where denominator = 0
- Even roots: radicand must be ≥ 0
- Logarithms: argument must be > 0
Interval notation: $(a, b)$ is open (excludes endpoints), $[a, b]$ is closed (includes endpoints).
Example 1 — Domain
Find the domain of $f(x) = \dfrac{\sqrt{x-3}}{x-7}$
Radicand: $x - 3 \geq 0 \implies x \geq 3$
Denominator: $x - 7 \neq 0 \implies x \neq 7$
Domain: $[3, 7) \cup (7, \infty)$
Transformation Rules
Given parent function $f(x)$:
| Transformation | Form | Effect |
|---|---|---|
| Vertical shift up | $f(x) + k$ | Up $k$ |
| Vertical shift down | $f(x) - k$ | Down $k$ |
| Horizontal shift right | $f(x - h)$ | Right $h$ |
| Horizontal shift left | $f(x + h)$ | Left $h$ |
| Vertical stretch | $a \cdot f(x)$, $ | a |
| Vertical compression | $a \cdot f(x)$, $0 < | a |
| Reflection over $x$-axis | $-f(x)$ | Flip vertically |
| Reflection over $y$-axis | $f(-x)$ | Flip horizontally |