Question

A new machine is purchased by a business for $200,000. For tax reasons, we assume the machine will lose $40,000 in value each year. If t is the number of years since the machine was purchased, and if V (t) represents the value of the machine after t years, write an equation in the form V (t)=mt+b for the value of the machine. Determine the domain and range for the function V (t).

Answer 1

Solution

Back-up Theory

Domain of a function y = f(x) is the set of values of x for which the function is defined. ........................................... (1)

Range of a function y = f(x) is the set of values of y for set of values of x for which the function is defined. ............ (2)

Now to work out the solution,

When t = 0, V(t) = b. but, at t = 0, the value of the machine is its initial value. So, we have: b = 200000 ................. (3)

‘we assume the machine will lose $40,000 in value each year.’ => at t = 1, value of the machine is: 200000 – 40000 = 160000. So, V(1) = m + b = 160000............................................................................................ (4)

(3) in (4): m + 200000 = 160000 or m = - 40000 ........................................................................................................(5)

(3) and (5) => equation for the value of the machine is: V(t) = - 40000t + 200000 Answer 1

Domain

Since t represents the time in years, it cannot be 0. So, minimum value t can take is 0. Again, value of the machine also cannot be negative. So, Answer 1 => 200000 – 40000t ≥ 0. Or, t ≤ 5

Thus, vide (1), domain of V(t) is: D = {t: t is an integer, 0 ≤ t ≤ 5}; i.e., Domain is: {0, 1, 2, 3, 4, 5} Answer 2

Range

Substituting the domain values in Answer 1, and vide (2), Range is: {200000, 160000, 120000, 80000, 40000, 0} Answer 3

DONE