Algebra and Functions
Core ideas
A candidate should be fluent with manipulating expressions (factoring, expanding, completing the square), solving linear and quadratic equations and inequalities, and understanding functions as rules with domain, range, composition, and inverses. Know the shapes and key features of linear, quadratic, polynomial, exponential, logarithmic, and absolute-value graphs, and how transformations (shifts, stretches, reflections) act on them.
Traps to watch
- Dividing by a variable expression and losing solutions; always check whether the divisor can be zero.
- Extraneous roots from squaring both sides of an equation.
- Domain restrictions: log arguments must be positive, denominators nonzero, even roots nonnegative.
- Inequality direction flipping when multiplying or dividing by a negative.
- Confusing f(g(x)) with g(f(x)); composition is not generally commutative.
Worked example
Solve x + 1 = sqrt(2x + 7). Square: x^2 + 2x + 1 = 2x + 7, so x^2 = 6, x = ±sqrt6. Check: x = sqrt6 gives LHS sqrt6 + 1 > 0 and RHS positive, works. x = -sqrt6 gives LHS 1 - sqrt6 < 0, but RHS is a nonnegative root, so reject. Solution set: {sqrt6}. The check step is the whole point.
Rule of thumb
Whenever you square, take an even root, or cancel a factor containing the variable, substitute your answers back into the original equation.