Worksheet: Rational Functions

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

Q1:

The figure shows the graph of 𝑦=1𝑥.

Write down the equations of the two asymptotes of 𝑦=1𝑥.

  • A 𝑦 = − 1 and 𝑥=−1
  • B 𝑦 = 1 and 𝑥=1
  • C 𝑦 = 1 and 𝑥=0
  • D 𝑦 = 0 and 𝑥=0
  • E 𝑦 = 0 and 𝑥=1

What is the domain of the function?

  • A 𝑥 ∈ ℝ , 𝑥 ≠ 0
  • B 𝑥 ∈ ( 1 , ∞ )
  • C 𝑥 ∈ ( 0 , ∞ )
  • D 𝑥 ∈ ℝ
  • E 𝑥 ∈ ( − ∞ , 0 )

What is the range of the function?

  • A 𝑦 ∈ ℝ
  • B 𝑦 ∈ ( 0 , ∞ )
  • C 𝑦 ∈ ( − ∞ , 0 )
  • D 𝑦 ∈ ℝ , 𝑦 ≠ 0
  • E 𝑦 ∈ ( 1 , ∞ )

Q2:

Consider the function 𝑦=3𝑥5𝑥+7.

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

  • A 𝑥 ∈ ℝ , 𝑥 ≠ 7 5
  • B 𝑥 ∈ ℝ , 𝑥 ≠ 5 7
  • C 𝑥 ∈ ℝ , 𝑥 ≠ 3 5
  • D 𝑥 ∈ ℝ , 𝑥 ≠ − 7 5
  • E 𝑥 ∈ ℝ , 𝑥 ≠ − 5 7

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

  • A 3 5 +
  • B 3 𝑥 5 +
  • C 3 5 + 7
  • D 3 5 𝑥 + 7

Now, taking the limit of this expression as 𝑥 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.

  • A 𝑦 ∈ ℝ , 𝑦 ≠ 1 4
  • B 𝑦 ∈ ℝ , 𝑦 ≠ 3 7
  • C 𝑦 ∈ ℝ , 𝑦 ≠ − 3 5
  • D 𝑦 ∈ ℝ , 𝑦 ≠ − 3 7
  • E 𝑦 ∈ ℝ , 𝑦 ≠ 3 5

Hence, state the equations of the two asymptotes.

  • A 𝑦 = 3 7 and 𝑥=75
  • B 𝑦 = 3 5 and 𝑥=−75
  • C 𝑦 = 3 7 and 𝑥=−75
  • D 𝑦 = 1 4 and 𝑥=35
  • E 𝑦 = 3 5 and 𝑥=75

Q3:

The following is the graph of the triangle wave function 𝑦=𝑔(𝑥).

What is the domain of its reciprocal function 𝑓(𝑥)=1𝑔(𝑥)?

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

Q4:

Find 𝑛(6) for the function 𝑛(𝑥)=3𝑥+7.

  • A − 3
  • B 1 2
  • C − 3 1 3
  • D 3 7
  • E 3 1 3

Q5:

Simplify the function 𝑛(𝑥)=(7𝑥−4)−(2𝑥+1)120𝑥−40, and find the values of 𝑥 for which (𝑛(𝑥))=16.

  • A 𝑛 ( 𝑥 ) = 2 4 ( 𝑥 + 1 ) , 𝑥 = − 5 6 or −76
  • B 𝑛 ( 𝑥 ) = 3 8 ( 𝑥 + 1 ) , 𝑥 = 2 9 3 or −353
  • C 𝑛 ( 𝑥 ) = 2 4 ( 𝑥 − 1 ) , 𝑥 = 7 6 or 56
  • D 𝑛 ( 𝑥 ) = 3 5 ( 𝑥 − 1 ) , 𝑥 = 2 3 3 or −173
  • E 𝑛 ( 𝑥 ) = 3 8 ( 𝑥 − 1 ) , 𝑥 = 3 5 3 or −293

Q6:

Given that 𝑛(𝑥)=7+𝑏𝑥−7, 𝑛(𝑥)=2𝑥−7, and 𝑛(𝑥)=𝑛(𝑥), what is the value of 𝑏?

Q7:

Given that 𝑓∶ℝ−{−1}→ℝ, where 𝑓(𝑥)=𝑥+𝑎𝑥−𝑏 and 𝑓(−5)=−14, determine the values of 𝑎 and 𝑏.

  • A 𝑎 = 1 , 𝑏 = − 2 1
  • B 𝑎 = − 9 2 , 𝑏 = 1
  • C 𝑎 = − 5 , 𝑏 = − 1
  • D 𝑎 = 6 , 𝑏 = − 1
  • E 𝑎 = − 1 , 𝑏 = − 2 9

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