Algebra (Subtest I)
Functions and Transformations
Algebra on Subtest I emphasizes structure: domain and range, composition, inverses, and families of functions (polynomial, rational, radical, exponential, logarithmic, trigonometric). A function has an inverse only if it is one-to-one; restrict the domain when necessary. For f(x) and g(x), (f∘g)(x) = f(g(x)), and the domain of the composition excludes inputs where g(x) is outside f's domain.
Polynomials and Equations
The Remainder Theorem says the remainder of p(x) divided by x − c is p(c); the Factor Theorem says x − c is a factor exactly when p(c) = 0. The Rational Root Theorem limits rational zeros to ±p/q where p divides the constant term and q divides the leading coefficient. The Fundamental Theorem of Algebra guarantees n complex roots counted with multiplicity for a degree-n polynomial; irrational and complex roots of real polynomials occur in conjugate pairs.
Exponential and Logarithmic Rules
Know log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(x^k) = k log_b x, and the change-of-base formula. Exponentials and logarithms are inverses, so b^{log_b x} = x for x > 0. Growth and decay models use A(t) = A₀e^{kt} or A₀(1+r)^t.
Worked Example
Solve log₂x + log₂(x − 2) = 3. Combine: log₂[x(x−2)] = 3, so x(x−2) = 8, giving x² − 2x − 8 = 0 and (x−4)(x+2) = 0. Candidates are x = 4 and x = −2. The domain requires x > 0 and x − 2 > 0, so x > 2; therefore x = −2 is extraneous and the only solution is x = 4. Checking confirms log₂4 + log₂2 = 2 + 1 = 3.
Traps That Cost Marks
- Skipping the domain check after combining or squaring, which admits extraneous roots.
- Writing log(x + y) = log x + log y; the sum rule applies to products only.
- Confusing arithmetic and geometric sequences: a_n = a₁ + (n−1)d versus a_n = a₁r^{n−1}.
- Misordering matrix multiplication when composing transformations.
- Forgetting that dividing by a variable can lose the solution x = 0.