Algebra and Functions
The Algebra and Functions domain tests function behavior, transformations, inverses, and the algebraic families that model change: polynomial, rational, exponential, and logarithmic.
Functions and inverses
A function assigns exactly one output per input. Its domain may be restricted by denominators or even roots. A one-to-one function has an inverse found by swapping x and y and solving; the inverse's domain equals the original function's range.
Transformations and families
For y = a·f(b(x − h)) + k, |a| is the vertical stretch, b compresses horizontally by 1/|b|, h shifts right, and k shifts up. Polynomial end behavior follows the leading term; a zero of even multiplicity touches the axis, odd multiplicity crosses it. Rational functions have vertical asymptotes where the reduced denominator is zero, and horizontal asymptotes determined by comparing degrees. Exponential growth has a constant ratio per unit; the logarithm is its inverse and has domain x > 0.
Worked example
Find the inverse of f(x) = (2x + 1)/(x − 3). Set y = (2x+1)/(x−3). Multiply: y(x − 3) = 2x + 1, so yx − 2x = 3y + 1, giving x(y − 2) = 3y + 1 and x = (3y + 1)/(y − 2). Thus f⁻¹(x) = (3x + 1)/(x − 2), with domain x ≠ 2, matching the range of f.
Traps
Forgetting to restrict a composite function's domain; accepting extraneous roots introduced by squaring; reading f⁻¹ as 1/f; misreading the sign of a horizontal shift inside the parentheses.