AP®MathsCollege BoardPrecalculusExam QuestionsPolynomial & Rational FunctionsRational Functions

Rational Functions (College Board AP® Precalculus): Exam Questions

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Rational Functions

1a
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2 marks

A drug is administered to a patient. The concentration of the drug in the patient's bloodstream, in milligrams per liter (mg/L), t hours after administration is modeled by the rational function

C open parentheses t close parentheses equals fraction numerator 50 t over denominator t squared plus 9 end fraction

for t greater or equal than 0.

Find the concentration of the drug in the patient's bloodstream 3 hours after administration. Include units in your answer.

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1b
2 marks

Determine limit as t rightwards arrow infinity of C open parentheses t close parentheses and interpret its meaning in the context of this problem.

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2
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2 marks

The function g is given by g open parentheses x close parentheses equals fraction numerator 2 x cubed minus 5 x plus 1 over denominator x minus 3 end fraction.

(i) Find all values of x, as decimal approximations, for which g open parentheses x close parentheses equals negative 4, or indicate that there are no such values.

(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.

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3a
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2 marks

The rational function space f is given by space f open parentheses x close parentheses equals fraction numerator x squared minus 3 x minus 10 over denominator x squared minus 25 end fraction .

Find all real zeros of space f open parentheses x close parentheses. Show the work that leads to your answer.

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3b
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2 marks

The graph of space f has a hole at x equals 5. Determine the space y-coordinate of the hole. Express your answer using the mathematical notation of a limit.

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3c
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2 marks

State the equation of the vertical asymptote of the graph of space f. Express the unbounded behavior of space f near this asymptote using a one-sided limit as x rightwards arrow negative 5 to the power of plus.

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4a
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1 mark

The rational function h is given by h open parentheses x close parentheses equals fraction numerator 2 x minus 8 over denominator x squared plus 2 x minus 3 end fraction .

Determine the domain of h.

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4b
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3 marks

Solve the inequality h open parentheses x close parentheses greater or equal than 0. Show the work that leads to your answer, including the use of a sign test.

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5a
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3 marks

The rational function g is given by g open parentheses x close parentheses equals fraction numerator x cubed minus 2 x squared plus 4 x minus 5 over denominator x squared plus 1 end fraction .

Use polynomial long division to rewrite g open parentheses x close parentheses in the equivalent form q open parentheses x close parentheses plus fraction numerator r open parentheses x close parentheses over denominator x squared plus 1 end fraction, where q open parentheses x close parentheses and r open parentheses x close parentheses are polynomials and the degree of r \left(x\right) is less than 2. Show your work.

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5b
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2 marks

Determine the equation of the slant asymptote of the graph of g. Justify your answer using the result from part (a) and the end behavior of the remainder term as x increases without bound.

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6a
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2 marks

A rational function space p is given in two equivalent analytic forms.

Standard form: space p open parentheses x close parentheses equals fraction numerator 3 x squared minus 12 x over denominator x squared minus 2 x minus 8 end fraction

Factored form: space p open parentheses x close parentheses equals fraction numerator 3 x open parentheses x minus 4 close parentheses over denominator open parentheses x minus 4 close parentheses open parentheses x plus 2 close parentheses end fraction

Determine the end behavior of space p as x increases without bound. Express your answer using the mathematical notation of a limit.

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6b
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1 mark

Identify the x-coordinate of any holes in the graph of space p.

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6c
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3 marks

Solve the equation space p open parentheses x close parentheses equals 2, or indicate that there are no such values. Show the work that leads to your answer.

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7a
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1 mark

Let r open parentheses x close parentheses equals fraction numerator a x squared plus b x plus c over denominator x squared plus d x minus 6 end fraction, where a, b, c and d are constants.

The graph of r has all of the following features:

  • a horizontal asymptote at y equals 2

  • a vertical asymptote at x equals 3

  • a hole at x equals negative 2

  • an x-intercept at x equals 1

Use the horizontal asymptote to find the value of a.

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7b
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2 marks

Use the vertical asymptote and the hole to find the value of d. Show the work that leads to your answer.

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7c
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3 marks

Use the x-intercept and the hole to find the values of b and c. Show the work that leads to your answer.

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