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The function π and π are defined by
π(π₯) =1/π₯, π₯ > 0 and π(π₯) = π^(βπ₯), π₯ β₯ 0
Prove that the function ππ exists (using domain & range) and deduce its range.[6 marks] -
A cable television company estimates that with π₯ thousand subscribers its monthly
revenue and cost (in thousands of dollars) are
π (π₯) = 32π₯ β 0.21π₯^(2) and πΆ(π₯) = 195 + 12π₯
(a) Sketch π
(π₯) and πΆ(π₯) in the same π₯ β π¦ axis. [6 marks] {Scale for x-axis: 0 β€ π₯ β€ 200, 0 β€ π¦ β€ 1400}
(b) Determine from both algebraically and graphically the companyβs break-even points; that is, find the number of subscribers at which the revenue equals the cost. [5 marks]
- We will examine the 2015 tax rates for a person. We use a piece-wised function, π to
denotes the tax rates, of income π₯, for π₯ β₯ 0
π(π₯) ={0.10π₯, if 0 β€ π₯ β€ 16050
1605 + 0.15(π₯ β 16050) if 16050 < π₯ β€ 65100
8962.50 + 0.25(π₯ β 65100) if 65100 < π₯ β€ 131450
25550 + 0.28(π₯ β 131450) if 131450 < π₯ β€ 200300
44828 + 0.33(π₯ β 200300) if 200,300 < π₯ β€ 357700
96770 + 0.35(π₯ β 357700) if π₯ > 357700
(a) Use the income tax function π 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 πΌ where it satisfies for any function π, for π β πΌ = π = πΌ β π [3 marks]
Β© Construct a function π where π = πΌ β π. The function π gives, for each pretax income π₯,
the amount that the taxpayer gets as take-home income and is, like π, a piece-wised
function. Write a complete description for π, in terms of cases, as we have done for π. [7 marks]
(d)Sketch the graph of the function π defined in 9(b). Make a short description about graph π.[10 marks]