Number and Quantity, Algebra, and Functions
Number and Quantity
What to know
This domain tests structure as much as computation: subsets of the real numbers (natural, whole, integer, rational, irrational), field and closure properties, and why numbers like √2 are irrational. Expect rational exponents and radicals (x^(m/n) = ⁿ√(x^m)), scientific notation, proportional reasoning, percent change, and unit conversion by dimensional analysis. Complex numbers appear routinely - conjugates, the fact that i² = −1, and modulus - and a simplified answer must be in a + bi form. Vectors (components, magnitude, direction, addition) and matrices (dimensions, multiplication conformability, determinant, inverse, non-commutativity) round out the domain, along with elementary number theory: primes, GCF and LCM, divisibility rules, and modular arithmetic.
Traps
- √(x²) = |x|, not x, and √(a + b) is not √a + √b.
- An even root of a negative number is not real; say so rather than forcing a negative output.
- A percent increase followed by the same percent decrease does not return to the original value, and percent change always divides by the original amount.
- AB ≠ BA for matrices, and only square matrices with nonzero determinant are invertible.
- Rationalizing a complex denominator requires multiplying by the full conjugate, not by the conjugate's terms separately.
Worked example
Simplify (3 + 2i)/(1 − i). Multiply numerator and denominator by the conjugate 1 + i. The denominator becomes (1 − i)(1 + i) = 1 − i² = 2. The numerator becomes 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i. So the result is 1/2 + (5/2)i.
Algebra and Functions
What to know
Function concepts dominate this domain: definition, function notation, domain and range, composition, inverses, and transformations of the form a·f(b(x − h)) + k. You should classify and graph linear, quadratic, polynomial, rational, exponential, logarithmic, radical, and absolute value functions; predict end behavior from degree and leading coefficient; find zeros and their multiplicities; locate vertical, horizontal, and slant asymptotes; and solve equations, inequalities, and systems. Sequences and series belong here too - arithmetic and geometric nth terms and sums - as does modeling: translating a verbal situation into an equation and interpreting slope, intercepts, and growth factors in context.
Traps
- The domain of a composition is not just the domain of the inner function; the outer function's restrictions apply to the output.
- f⁻¹(x) means the inverse function, not 1/f(x).
- log(ab) = log a + log b, but log(a + b) is not log a + log b, and the argument must be positive.
- A rational function may cross its horizontal asymptote; only vertical asymptotes are excluded from the domain.
- A zero of even multiplicity touches the axis without crossing it.
- Multiplying or dividing an inequality by a negative reverses the direction.
Worked example
Find the inverse of f(x) = 2x − 5. Write y = 2x − 5, swap variables to get x = 2y − 5, and solve: y = (x + 5)/2. Verify by composing: f(f⁻¹(x)) = 2·[(x + 5)/2] − 5 = x. Rule of thumb: never trust an inverse until both compositions return x.