KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 02 · Lesson 08

Optimization and Technology-Assisted Validation

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Optimization and Technology-Assisted Validation

U02 · L08
1

State the domain of f(x)=(x+1)/(x−4).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: x≠4
Reasoning: The denominator cannot equal zero.
2

Find the vertical asymptote of f(x)=3/(x+2).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: x=−2
Reasoning: The noncanceled denominator is zero at −2.
3

Find the horizontal asymptote of (2x²+1)/(x²−9).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: y=2
Reasoning: Equal degrees: ratio of leading coefficients is 2.
4

Simplify (x²−9)/(x−3) and identify the excluded value.

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: x+3, with x≠3
Reasoning: Factor and cancel, retaining the original restriction.
5

Solve 1/x+1/(x+2)=1.

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: x=±√2
Reasoning: Multiply by x(x+2), solve x²=2, and verify restrictions.
6

On which interval is 1/(x−5) positive?

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
Answer: (5,∞)
Reasoning: The numerator is positive and denominator is positive when x>5.
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