Lesson Worksheet: Rational Functions Mathematics • 10th Grade

In this worksheet, we will practice identifying, writing, and evaluating a rational function.


The figure shows the graph of 𝑦=1π‘₯.

Write down the equations of the two asymptotes of 𝑦=1π‘₯.

  • A𝑦=βˆ’1 and π‘₯=βˆ’1
  • B𝑦=1 and π‘₯=1
  • C𝑦=1 and π‘₯=0
  • D𝑦=0 and π‘₯=0
  • E𝑦=0 and π‘₯=1

What is the domain of the function?

  • Aπ‘₯βˆˆβ„,π‘₯β‰ 0
  • Bπ‘₯∈(1,∞)
  • Cπ‘₯∈(0,∞)
  • Dπ‘₯βˆˆβ„
  • Eπ‘₯∈(βˆ’βˆž,0)

What is the range of the function?

  • Aπ‘¦βˆˆβ„
  • Bπ‘¦βˆˆ(0,∞)
  • Cπ‘¦βˆˆ(βˆ’βˆž,0)
  • Dπ‘¦βˆˆβ„,𝑦≠0
  • Eπ‘¦βˆˆ(1,∞)


Consider the function 𝑦=3π‘₯5π‘₯+7.

By considering the point at which the denominator equals zero, find the domain of the function.

  • Aπ‘₯βˆˆβ„,π‘₯β‰ 75
  • Bπ‘₯βˆˆβ„,π‘₯β‰ 57
  • Cπ‘₯βˆˆβ„,π‘₯β‰ 35
  • Dπ‘₯βˆˆβ„,π‘₯β‰ βˆ’75
  • Eπ‘₯βˆˆβ„,π‘₯β‰ βˆ’57

To find the range of the function, a handy trick is to divide the numerator and denominator of 3π‘₯5π‘₯+7 through by π‘₯. What expression does this give us?

  • A35+οŠ­ο—
  • B3π‘₯5+οŠ­ο—
  • C35+7
  • D35π‘₯+7

Now, taking the limit of this expression as π‘₯ tends to infinity will give us the value of 𝑦 which is not in the range of the original function. Use this to state the range of the function.

  • Aπ‘¦βˆˆβ„,𝑦≠14
  • Bπ‘¦βˆˆβ„,𝑦≠37
  • Cπ‘¦βˆˆβ„,π‘¦β‰ βˆ’35
  • Dπ‘¦βˆˆβ„,π‘¦β‰ βˆ’37
  • Eπ‘¦βˆˆβ„,𝑦≠35

Hence, state the equations of the two asymptotes.

  • A𝑦=37 and π‘₯=75
  • B𝑦=35 and π‘₯=βˆ’75
  • C𝑦=37 and π‘₯=βˆ’75
  • D𝑦=14 and π‘₯=35
  • E𝑦=35 and π‘₯=75


The following is the graph of the triangle wave function 𝑦=𝑔(π‘₯).

What is the domain of its reciprocal function 𝑓(π‘₯)=1𝑔(π‘₯)?

  • Aall real numbers
  • Beven integers
  • Call integers
  • Dall real numbers that are not integers
  • Eodd integers


Find 𝑛(6) for the function 𝑛(π‘₯)=3π‘₯+7.

  • Aβˆ’3
  • B12
  • Cβˆ’313
  • D37
  • E313


Simplify the function 𝑛(π‘₯)=(7π‘₯βˆ’4)βˆ’(2π‘₯+1)120π‘₯βˆ’40, and find the values of π‘₯ for which (𝑛(π‘₯))=16.

  • A𝑛(π‘₯)=24(π‘₯+1), π‘₯=βˆ’56 or βˆ’76
  • B𝑛(π‘₯)=38(π‘₯+1), π‘₯=293 or βˆ’353
  • C𝑛(π‘₯)=24(π‘₯βˆ’1), π‘₯=76 or 56
  • D𝑛(π‘₯)=35(π‘₯βˆ’1), π‘₯=233 or βˆ’173
  • E𝑛(π‘₯)=38(π‘₯βˆ’1), π‘₯=353 or βˆ’293


Given that 𝑛(π‘₯)=7+𝑏π‘₯βˆ’7, 𝑛(π‘₯)=2π‘₯βˆ’7, and 𝑛(π‘₯)=𝑛(π‘₯), what is the value of 𝑏?


Given that π‘“βˆΆβ„βˆ’{βˆ’1}→ℝ, where 𝑓(π‘₯)=π‘₯+π‘Žπ‘₯βˆ’π‘ and 𝑓(βˆ’5)=βˆ’14, determine the values of π‘Ž and 𝑏.

  • Aπ‘Ž=1, 𝑏=βˆ’21
  • Bπ‘Ž=βˆ’92, 𝑏=1
  • Cπ‘Ž=βˆ’5, 𝑏=βˆ’1
  • Dπ‘Ž=6, 𝑏=βˆ’1
  • Eπ‘Ž=βˆ’1, 𝑏=βˆ’29


The function 𝑛(π‘₯) has two asymptotes at 𝑦=53 and π‘₯=4. Given that 𝑛(π‘₯)=π‘Žπ‘₯+33π‘₯βˆ’π‘, determine the values of π‘Ž and 𝑏.

  • Aπ‘Ž=5, 𝑏=233
  • Bπ‘Ž=12, 𝑏=5
  • Cπ‘Ž=12, 𝑏=1
  • Dπ‘Ž=5, 𝑏=βˆ’12
  • Eπ‘Ž=5, 𝑏=12


Write a rational function in the simplest form 𝑛(π‘₯)=π‘Žπ‘₯+𝑏𝑐π‘₯+𝑑, given that the vertical asymptote is at π‘₯=βˆ’3, the horizontal asymptote is at 𝑦=2, and 𝑛(2)=1.

  • A𝑛(π‘₯)=π‘₯βˆ’3π‘₯+3
  • B𝑛(π‘₯)=2π‘₯+6π‘₯+3
  • C𝑛(π‘₯)=βˆ’3π‘₯+10π‘₯+2
  • D𝑛(π‘₯)=2π‘₯βˆ’5π‘₯βˆ’3
  • E𝑛(π‘₯)=2π‘₯+1π‘₯+3


Find the multiplicative inverse of 𝑛(π‘₯)=5π‘₯βˆ’32π‘₯.

  • A2π‘₯10π‘₯βˆ’3
  • B2π‘₯5π‘₯βˆ’3
  • C15π‘₯βˆ’2π‘₯3
  • D10π‘₯βˆ’32π‘₯
  • E5π‘₯βˆ’32π‘₯

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