GACE Mathematics (522) cheat sheet

State domain restrictions before simplifying; cancelled factors still exclude values.

Composition f(g(x)) is evaluated inside-out and is generally not commutative.

Inverses require one-to-one functions; domain and range swap.

Horizontal transformations act opposite to the sign or factor shown.

Log laws convert sums to products, enabling solution of log equations.

Always check candidate solutions against original domain restrictions.

Rational numbers are ratios of integers; square roots of non-perfect squares and pi are irrational.

Exponent laws apply to multiplication, division, and powers of powers, but never distribute across addition.

Percent change uses the original value as the base; successive percent changes multiply rather than add.

GCF uses shared primes at their lowest powers; LCM uses all primes at their highest powers.

Scientific notation requires a coefficient between 1 and 10; check the magnitude to catch exponent errors.

Similarity scales lengths by k, areas by k², volumes by k³.

Inscribed angle equals half the intercepted central arc.

Sine law requires a known side–opposite-angle pair; watch the ambiguous case.

Cosine law handles SAS and SSS configurations.

Arc length and sector area formulas require radians.

Pythagorean identity sin²θ + cos²θ = 1 underpins most simplifications.

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

An implication and its contrapositive are equivalent; the converse is not.

Negating a quantified statement flips the quantifier and negates the predicate.

Induction needs a base case, a stated hypothesis, and the step.

Inclusion–exclusion corrects for double counting.

Pigeonhole principle forces a collision when items exceed categories.

Sum of vertex degrees equals twice the edge count.

Median resists outliers; mean does not.

P(A ∪ B) subtracts the intersection to avoid double counting.

Conditional probability divides the joint probability by the condition's probability.

Combinations count selections; permutations count arrangements.

z = (x − μ)/σ places a value on the standard normal scale.

Random assignment, not random sampling alone, supports causal conclusions.