Algebra and Functions
Algebra and Functions
This domain tests function behavior-domain, range, inverses, transformations, rates of change-plus polynomial, rational, exponential, logarithmic, and trigonometric families, sequences and series, and systems of equations. Expect questions that ask what a function does, not just what its equation is.
Core rules and concepts
- A function is a rule assigning exactly one output per input; domain restrictions come from denominators, even roots, and log arguments.
- Transformations: f(x − h) shifts right h; af(x) scales vertically; f(−x) reflects over the y-axis.
- Inverses: swap x and y, then solve; the inverse's domain is the original's range.
- Logarithms: log_b(xy) = log_b x + log_b y, log_b(x^y) = y log_b x, and log_b 1 = 0.
- Exponential growth/decay: f(x) = a·b^x with b = 1 + r (growth) or 1 − r (decay).
Worked example
Find the inverse of f(x) = 2^(x−1) + 3. Write y = 2^(x−1) + 3. Subtract 3: y − 3 = 2^(x−1). Take log base 2: log₂(y − 3) = x − 1. Add 1: x = log₂(y − 3) + 1. So f⁻¹(x) = log₂(x − 3) + 1, with domain x > 3. Reasoning: the horizontal shift becomes a vertical shift after swapping, and the range y > 3 of f becomes the domain of f⁻¹.
Traps that cost marks
- Forgetting to restrict the domain of an inverse.
- Applying log properties to sums: log(a + b) ≠ log a + log b.
- Introducing extraneous solutions when squaring both sides.
- Mixing up growth factor b with rate r (10% growth → b = 1.10, not 0.10).