Unit 2
Two families defined by what they exclude. Radicals restrict the domain; rationals introduce asymptotes and holes, and both create extraneous solutions.
Lesson 2.1
Isolate a radical before raising both sides to a power. Squaring can introduce solutions that do not satisfy the original equation. Record the original real-domain restrictions and check every candidate in the equation before reporting it.
Lesson 2.2
Factor numerator and denominator before describing a rational graph. Every original denominator zero is excluded. A canceled factor can produce a hole, while a remaining denominator zero typically produces a vertical asymptote after the expression is reduced. Cancellation does not restore an excluded input.
Lesson 2.3
Compare numerator and denominator degrees to determine rational end behavior. A smaller numerator degree gives horizontal asymptote y = 0. Equal degrees give the ratio of leading coefficients. A numerator degree exactly one larger can give a slant asymptote found by division.
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