Maths Assignment 2016

  1. The function 𝑓 and 𝑔 are defined by
    𝑓(π‘₯) =1/π‘₯, π‘₯ > 0 and 𝑔(π‘₯) = 𝑒^(βˆ’π‘₯), π‘₯ β‰₯ 0
    Prove that the function 𝑓𝑔 exists (using domain & range) and deduce its range.[6 marks]

  2. A cable television company estimates that with π‘₯ thousand subscribers its monthly
    revenue and cost (in thousands of dollars) are
    𝑅(π‘₯) = 32π‘₯ βˆ’ 0.21π‘₯^(2) and 𝐢(π‘₯) = 195 + 12π‘₯

(a) Sketch 𝑅(π‘₯) and 𝐢(π‘₯) in the same π‘₯ βˆ’ 𝑦 axis. [6 marks] {Scale for x-axis: 0 ≀ π‘₯ ≀ 200, 0 ≀ 𝑦 ≀ 1400}
(b) Determine from both algebraically and graphically the company’s break-even points; that is, find the number of subscribers at which the revenue equals the cost. [5 marks]

  1. We will examine the 2015 tax rates for a person. We use a piece-wised function, 𝑓 to
    denotes the tax rates, of income π‘₯, for π‘₯ β‰₯ 0
    𝑓(π‘₯) ={0.10π‘₯, if 0 ≀ π‘₯ ≀ 16050
    1605 + 0.15(π‘₯ βˆ’ 16050) if 16050 < π‘₯ ≀ 65100
    8962.50 + 0.25(π‘₯ βˆ’ 65100) if 65100 < π‘₯ ≀ 131450
    25550 + 0.28(π‘₯ βˆ’ 131450) if 131450 < π‘₯ ≀ 200300
    44828 + 0.33(π‘₯ βˆ’ 200300) if 200,300 < π‘₯ ≀ 357700
    96770 + 0.35(π‘₯ βˆ’ 357700) if π‘₯ > 357700

(a) Use the income tax function 𝑓 to determine the tax on the given taxable income in the
year 2015.
(i) RM27000 [2 marks]
(ii) RM89000 [2 marks]
(iii) RM350000 [2 marks]
(iv) RM560000 [2 marks]

(b) Find function 𝐼 where it satisfies for any function 𝑓, for 𝑓 ∘ 𝐼 = 𝑓 = 𝐼 ∘ 𝑓 [3 marks]

Β© Construct a function 𝑔 where 𝑔 = 𝐼 βˆ’ 𝑓. The function 𝑔 gives, for each pretax income π‘₯,
the amount that the taxpayer gets as take-home income and is, like 𝑓, a piece-wised
function. Write a complete description for 𝑔, in terms of cases, as we have done for 𝑓. [7 marks]

(d)Sketch the graph of the function 𝑔 defined in 9(b). Make a short description about graph 𝑔.[10 marks]

Hello, why did you post the math assignment question here? Is it your college math course assignment? What exactly do you need help for?