Worksheet: Reciprocal Function Characteristics

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In this worksheet, we will practice identifying reciprocal functions and their properties and identifying asymptotes, domains, and ranges.

Q1:

The figure shows the graph of 𝑦 = 1 𝑥 .

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

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

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 ∈ ] − ∞ , 0 [
  • B 𝑦 ∈ ℝ
  • C 𝑦 ∈ ℝ , 𝑦 ≠ 0
  • D 𝑦 ∈ ] 0 , ∞ [
  • E 𝑦 ∈ ] 1 , ∞ [

Q2:

Consider the function 𝑦 = 3 𝑥 5 𝑥 + 7 .

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

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

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

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

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

Hence, state the equations of the two asymptotes.

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

Q3:

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

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

  • Aeven integers
  • Ball real numbers
  • Codd integers
  • Dall real numbers that are not integers
  • Eall integers

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