Algebra and Functions
What this domain tests
Algebra and function practice asks you to move fluently between symbolic, graphical, and verbal representations of relationships. Expect manipulation of polynomial, rational, radical, exponential, and logarithmic expressions; solving equations and inequalities; and reasoning about domain, range, inverse functions, and transformations. Questions often present a function in one form and require an answer in another, so translation skill matters as much as raw computation.
Core rules and concepts
- Domain restrictions come first. Denominators cannot be zero; even-index radicals require nonnegative radicands; logarithm arguments must be positive. State restrictions before simplifying rational expressions, because cancelled factors still exclude original zeros.
- Function composition is not commutative. f(g(x)) and g(f(x)) generally differ; compute inside-out.
- Inverses undo. f⁻¹ exists only if f is one-to-one on its stated domain; swap x and y, then solve. Domains and ranges swap too.
- Transformations: af(b(x−h))+k gives vertical stretch by |a|, horizontal compression by 1/|b|, shift right h, up k. Horizontal changes behave opposite to intuition.
- Exponential vs. logarithmic: y = bˣ and y = log_b x are inverses; log_b(MN) = log_b M + log_b N, log_b(M/N) = log_b M − log_b N, log_b(Mᵖ) = p log_b M.
- Rational functions: vertical asymptotes at non-cancelled denominator zeros, holes at cancelled ones, and end behavior from degree comparison.
Worked example
Solve log₂(x) + log₂(x − 2) = 3.
Combine: log₂[x(x − 2)] = 3, so x(x − 2) = 2³ = 8. Then x² − 2x − 8 = 0, giving (x − 4)(x + 2) = 0, so x = 4 or x = −2. Now check the domain: both log arguments must be positive, so x > 0 and x − 2 > 0, meaning x > 2. Therefore x = −2 is extraneous and the only solution is x = 4. Verification: log₂4 + log₂2 = 2 + 1 = 3. ✓
Traps that cost marks
- Reporting extraneous roots from squaring or from log equations without domain checking.
- Cancelling a factor in a rational expression and then claiming the cancelled value is in the domain.
- Applying a horizontal shift in the wrong direction, or stretching horizontally when the coefficient is inside the argument.
- Forgetting that log_b x is undefined for x ≤ 0 and that b must be positive and not 1.
- Losing the ± when solving x² = k.