stepr Draw the raps ff x stickers are 000002x dollars campai

stepr Draw the raps ff x stickers are .000002x dollars campaigns f Michers bumper sticker is P)- 0.1s-0 makes bumper ocaooy, then the prike per producing the order i -0.09 tem Number of items functions of x. to express Rikd the revenue from an order of x stickers, as a product of two What is the maximum revenue? What is the maximum number of stickers ordered? 3. What is the domain and range of Px)? What is the domain of R(x)? 4. Find the inverse of Ptx)·(You can think the function as y-t(x)·0.15-0000002x) What does f \'represent? (don\'t answer as it is the inverse, answer within the problem in terms of number of stickers ordered and price) 5. 6. What is the domain of f? What is the range of f? 7. What is the domain of f\'? What is the range of f\'? 8. Find f(0.10). What does your answer represent?

Solution

given x<100000, P(x)=0.15-0.000002x ,C(x)=0.095x-0.0000005x²

1)

revenue ,R(x) =(P(x))x

R(x) =(0.15-0.000002x)x

R(x) =(0.15x-0.000002x²)

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2)

R\'(x) =(0.15-0.000004x)

R\'\'(x) =0.15-0.000004

for critical points R\'(x)=0

=>(0.15-0.000004x)=0

=>x=37500

R(0)=0

R(37500)=2812.5

R(100000)=-5000

maximum revenue is 2812.5 dollars

maximum number of stickers ordered for maximum revenue =37500

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3)

P(x)0 , x<100000

=>0.15-0.000002x 0

=>0.150.000002x

=>x75000

domain of P(x) is [0,75000]

range of P(x) is  [0,0.15]

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4)

let P(x)=y=f(x)=0.15-0.000002x

=>x=f^-1(y)

0.15-0.000002x=y

=>0.000002x=0.15-y

=>x=500000(0.15-y)

=>f^-1(y)=500000(0.15-y)

inverse of P(x) is =500000(0.15-y)

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5)

f^-1 represents the number of tickets ordered given the price per ticket

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6)

domain of f is [0,75000]

range of f is  [0,0.15]

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7)

domain of f^-1 is [0,0.15]

range of f^-1 is [0,75000]

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8)

f^-1(0.10)=500000(0.15-0.10)

f^-1(0.10)=25000

25000 tickets are ordered when price per ticket is 0.10dollars