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,𝑥0
  • B𝑥(1,)
  • C𝑥(0,)
  • D𝑥
  • E𝑥(,0)

What is the range of the function?

  • A𝑦
  • B𝑦(0,)
  • C𝑦(,0)
  • D𝑦,𝑦0
  • 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,𝑥75
  • B𝑥,𝑥57
  • C𝑥,𝑥35
  • D𝑥,𝑥75
  • E𝑥,𝑥57

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,𝑦14
  • B𝑦,𝑦37
  • C𝑦,𝑦35
  • D𝑦,𝑦37
  • E𝑦,𝑦35

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.

  • A3
  • B12
  • C313
  • D37
  • E313

Q5:

Simplify the function 𝑛(𝑥)=(7𝑥4)(2𝑥+1)120𝑥40, 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(𝑥1), 𝑥=76 or 56
  • D𝑛(𝑥)=35(𝑥1), 𝑥=233 or 173
  • E𝑛(𝑥)=38(𝑥1), 𝑥=353 or 293

Q6:

Given that 𝑛(𝑥)=7+𝑏𝑥7, 𝑛(𝑥)=2𝑥7, and 𝑛(𝑥)=𝑛(𝑥), what is the value of 𝑏?

Q7:

Given that 𝑓{1}, where 𝑓(𝑥)=𝑥+𝑎𝑥𝑏 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+33𝑥𝑏, 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)=𝑥3𝑥+3
  • B𝑛(𝑥)=2𝑥+6𝑥+3
  • C𝑛(𝑥)=3𝑥+10𝑥+2
  • D𝑛(𝑥)=2𝑥5𝑥3
  • E𝑛(𝑥)=2𝑥+1𝑥+3

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