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.

- A reciprocal function is defined as

x |→ 1/x , x ∈**R**\ {0}. - 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}. - Answer the following questions with reference to your sketch above:
- Why is f(x) not defined at x = 0?
- What happen to the value of f(x) when x approaches positive infinity?
- What happen to the value of f(x) when x approaches zero and x > 0?
- What happen to the value of f(x) when x approaches negative infinity?
- What happen to the value of f(x) when x approaches zero and x < 0?
- Identify the vertical asymptote?____________________
- Identify the horizontal asymptote? __________________

- Find the inverse of f(x) = 1/x , x ∈
**R**\ {0}. - Identify both the domain and range of f
^{ -1}(x). - Compare the inverse of f(x) with f(x)?
- 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)]
- 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.]
- 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.]
- 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. - 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.