Worksheet: Rational Functions

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.

  • A𝑦=−1 and 𝑥=−1
  • B𝑦=1 and 𝑥=1
  • C𝑦=1 and 𝑥=0
  • D𝑦=0 and 𝑥=0
  • E𝑦=0 and 𝑥=1

What is the domain of the function?

  • A𝑥∈ℝ,𝑥≠0
  • B𝑥∈(1,∞)
  • C𝑥∈(0,∞)
  • D𝑥∈ℝ
  • E𝑥∈(−∞,0)

What is the range of the function?

  • A𝑦∈ℝ
  • B𝑦∈(0,∞)
  • C𝑦∈(−∞,0)
  • D𝑦∈ℝ,𝑦≠0
  • E𝑦∈(1,∞)

Q2:

Consider the function 𝑦=3𝑥5𝑥+7.

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

  • A𝑥∈ℝ,𝑥≠75
  • B𝑥∈ℝ,𝑥≠57
  • C𝑥∈ℝ,𝑥≠35
  • D𝑥∈ℝ,𝑥≠−75
  • E𝑥∈ℝ,𝑥≠−57

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?

  • A35+
  • B3𝑥5+
  • C35+7
  • D35𝑥+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.

  • A𝑦∈ℝ,𝑦≠14
  • B𝑦∈ℝ,𝑦≠37
  • C𝑦∈ℝ,𝑦≠−35
  • D𝑦∈ℝ,𝑦≠−37
  • E𝑦∈ℝ,𝑦≠35

Hence, state the equations of the two asymptotes.

  • A𝑦=37 and 𝑥=75
  • B𝑦=35 and 𝑥=−75
  • C𝑦=37 and 𝑥=−75
  • D𝑦=14 and 𝑥=35
  • E𝑦=35 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
  • B12
  • C−313
  • D37
  • E313

Q5:

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

  • A𝑛(𝑥)=24(𝑥+1), 𝑥=−56 or −76
  • B𝑛(𝑥)=38(𝑥+1), 𝑥=293 or −353
  • C𝑛(𝑥)=24(𝑥−1), 𝑥=76 or 56
  • D𝑛(𝑥)=35(𝑥−1), 𝑥=233 or −173
  • E𝑛(𝑥)=38(𝑥−1), 𝑥=353 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.

  • A𝑎=1, 𝑏=−21
  • B𝑎=−92, 𝑏=1
  • C𝑎=−5, 𝑏=−1
  • D𝑎=6, 𝑏=−1
  • E𝑎=−1, 𝑏=−29

Q8:

The function 𝑛(𝑥) has two asymptotes at 𝑦=53 and 𝑥=4. Given that 𝑛(𝑥)=𝑎𝑥+33𝑥−𝑏, determine the values of 𝑎 and 𝑏.

  • A𝑎=5, 𝑏=233
  • B𝑎=12, 𝑏=5
  • C𝑎=12, 𝑏=1
  • D𝑎=5, 𝑏=−12
  • E𝑎=5, 𝑏=12

Q9:

Write a rational function in the simplest form 𝑛(𝑥)=𝑎𝑥+𝑏𝑐𝑥+𝑑, given that the vertical asymptote is at 𝑥=−3, the horizontal asymptote is at 𝑦=2, and 𝑛(2)=1.

  • A𝑛(𝑥)=𝑥−3𝑥+3
  • B𝑛(𝑥)=2𝑥+6𝑥+3
  • C𝑛(𝑥)=−3𝑥+10𝑥+2
  • D𝑛(𝑥)=2𝑥−5𝑥−3
  • E𝑛(𝑥)=2𝑥+1𝑥+3

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