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Worksheet: Intro to Asymptotes of Rational Functions

Q1:

What are the two asymptotes of the hyperbola 𝑦 = 8 4 π‘₯ βˆ’ 3 + 5 3 ?

  • A π‘₯ = 3 4 , 𝑦 = 3 5
  • B π‘₯ = 4 3 , 𝑦 = 5 3
  • C π‘₯ = 1 4 , 𝑦 = 5 3
  • D π‘₯ = 3 4 , 𝑦 = 5 3
  • E π‘₯ = 1 4 , 𝑦 = 1 3

Q2:

Determine the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = βˆ’ 1 + 3 π‘₯ βˆ’ 4 π‘₯ 2 .

  • A The vertical asymptote is at π‘₯ = 3 , and the horizontal asymptote is at 𝑦 = βˆ’ 4 .
  • B The vertical asymptote is at 𝑦 = βˆ’ 1 , and the horizontal asymptote is at π‘₯ = 0 .
  • C The function has no vertical asymptote, and the horizontal asymptote is at 𝑦 = 0 .
  • D The vertical asymptote is at π‘₯ = 0 , and the horizontal asymptote is at 𝑦 = βˆ’ 1 .
  • E The vertical asymptote is at π‘₯ = βˆ’ 1 , and the function has no horizontal asymptote.

Q3:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 1 π‘₯ βˆ’ 5 in ℝ .

  • A The domain is { 0 } , and the range is ℝ .
  • B The domain is ℝ , and the range is ℝ .
  • C The domain is ℝ , and the range is { 0 } .
  • D The domain is ℝ βˆ’ { 0 } , and the range is ℝ βˆ’ { βˆ’ 5 } .

Q4:

What function is represented in the figure below?

  • A 𝑓 ( π‘₯ ) = 1 π‘₯ βˆ’ 3
  • B 𝑓 ( π‘₯ ) = 1 π‘₯ βˆ’ 3
  • C 𝑓 ( π‘₯ ) = βˆ’ 1 π‘₯ βˆ’ 3
  • D 𝑓 ( π‘₯ ) = βˆ’ 1 π‘₯ βˆ’ 3

Q5:

Find the domain of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 3 6 π‘₯ π‘₯ + 6 π‘₯ 3 2 .

  • A ℝ βˆ’ [ βˆ’ 6 , 0 ]
  • B ℝ βˆ’ { βˆ’ 1 2 , βˆ’ 6 }
  • C { βˆ’ 6 , 0 }
  • D ℝ βˆ’ { βˆ’ 6 , 0 }

Q6:

What function is represented in the figure below?

  • A 𝑓 ( π‘₯ ) = 1 π‘₯ + 3 + 2
  • B 𝑓 ( π‘₯ ) = 1 π‘₯ + 2 + 3
  • C 𝑓 ( π‘₯ ) = βˆ’ 1 π‘₯ + 3 + 2
  • D 𝑓 ( π‘₯ ) = βˆ’ 1 π‘₯ + 2 + 3

Q7:

Determine the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = 4 + 2 π‘₯ βˆ’ 3 π‘₯ 2 .

  • A The vertical asymptote is at π‘₯ = 2 , and the horizontal asymptote is at 𝑦 = βˆ’ 3 .
  • B The vertical asymptote is at 𝑦 = 4 , and the horizontal asymptote is at π‘₯ = 0 .
  • C The function has no vertical asymptote, and the horizontal asymptote is at 𝑦 = 0 .
  • D The vertical asymptote is at π‘₯ = 0 , and the horizontal asymptote is at 𝑦 = 4 .
  • E The vertical asymptote is at π‘₯ = 4 , and the function has no horizontal asymptote.