Reciprocal Function

Aims: By the end of this chapter, you will be able to
(i) define a reciprocal function, and
(ii) manipulate its properties to solve problems.

  1. A reciprocal function is defined as
    x |→ 1/x , x ∈R \ {0}.
  2. Example: The time (t) taken for a train to travel a fixed distance(D) against its speed (s). t = D/s or f(s)= D/s where D is a constant.

    Use your calculator to sketch f(x) = 1/x , x ∈ [-5, 5] \ {0}.
  3. Answer the following questions with reference to your sketch above:
    1. Why is f(x) not defined at x = 0?

    2. What happen to the value of f(x) when x approaches positive infinity?

    3. What happen to the value of f(x) when x approaches zero and x > 0?

    4. What happen to the value of f(x) when x approaches negative infinity?

    5. What happen to the value of f(x) when x approaches zero and x < 0?

    6. Identify the vertical asymptote?____________________
    7. Identify the horizontal asymptote? __________________

  4. Find the inverse of f(x) = 1/x , x ∈R \ {0}.
  5. Identify both the domain and range of f -1(x).

  6. Compare the inverse of f(x) with f(x)?

  7. With the help of your calculator sketch g(x) = 4/x, x ∈[-5, 5] \ {0} on the same axes as f(x) above. [Use a different colour for g(x)]

  8. How would a function h(x) = (k/x), x ∈[-5, 5] \ {0} and 0 < k < 1 look like? [Use a different colour and illustrate your answer on the same axes as f(x) above.]

  9. How would a function j(x) = (k/x), x ∈[-5, 5] \ {0} and k < -1 look like? [Use a different colour and illustrate your answer on the same axes as f(x) above.]

  10. Let f(x) = 1 + (2/x), x ∈R \ {0} . Solve f(x) = 0. Solution: 1 + (2/x) = 0
    2/x = -1
    2 = -x , since x is Not 0
    x = -2.

  11. Let f(x)= 3/(x + 2). What is the value of x when f(x) is not defined? Identify the horizontal asymptote of f(x). Solution: x = -2 . This is the vertical asymptote.
    As x approaches positive infinity, then we have 3/(a very big number). Thus, f(x) approaches 0. y = 0 is one horizontal asymptote.
    As x approaches negative infinity, then we have 3/(a very small negative number). Thus, f(x) approaches 0. y = 0 and this is the same horizontal asymptote as above.
    Domain: all real numbers except -2. Range: f(x)> 0 and f(x)< 0 or all real numbers except 0. Confirm these asymptotes using your calculator.

Exercises