OPERATIONS ON FUNCTIONS
1a. The research and development department of an automobile manufacturer has
determined that when required to stop quickly to avoid an accident, the distance (in feet) a
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car travels during the driver’s reaction time is given by R  x   x , where x is the speed of
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the car in miles per hour. The distance (in feet) traveled while the driver is braking is given
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by B  x   x 2 . Find the function that represents the total stopping distance T.
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1b. Use a graphing utility to graph the functions R, B, and T in the same viewing window for
0≤x≤60.
1c. Which function contributes most to the magnitude of the sum at higher speeds? Explain.
2a. There are multiple scales on which to measure temperature. The degree Fahrenheit (◦F)
scale is mainly used in the United States. Using this scale, water at sea level will boil at
212◦F and freeze at 32◦F. Another scale, the degree Celsuis (◦C) was devised so that water
at sea level will boil at 100◦C and freeze at 0◦C. This scale is used in science and is one of
the standards of the “metric” system of measurements. Given that the relationship
between the two scales is linear. Write a linear function f(x), to convert degrees Celsius (x
is the variable and represents temperature in degrees Celsius) to degrees Fahrenheit.
2b. Now find f-1(x). What does the inverse represent?
2c. The Kelvin (K) temperature scales measures temperature in kelvins (not degrees kelvin and
notice the lower case k). The abbreviation for kelvin is K. The Kelvin scale shifts the Celsius
scale down 273.15 so that the zero point is equal to absolute zero (a hypothetical
temperature at which there is a complete absence of heat energy). The formula to convert
from kelvins to Celsius c  x   x  273.15 , where x is the temperature in kelvins.
Find  f c  x  , explain what it represents.
3a. Suppose that you want to build a SQUARE fenced-in area for your dog. Fencing is
purchased in linear feet. Write an expression to describe side-length s(x), as a function of the
amount of fencing purchased x.
3b. Write an expression to describe area of a square A(x), based on side length x.
3c. Write a composite function that determines the area of your dog pen as a function of how
many linear feet are purchased.
3d. If you purchase 100 linear feet, what is the area of your dog pen?
3e. If you purchase 200 linear feet, what is the area of your dog pen?
4a. Suppose that you want to build a CIRCULAR fenced-in area for your dog. Fencing is
purchased in linear feet. Write a composite function that determines the area of your dog pen
as a function of how many linear feet are purchased.
4b. If you purchase 100 linear feet, what is the area of your dog pen?
4c. If you purchase 200 linear feet, what is the area of your dog pen?
5. Typical supply and demand relationships state that as the number of units for sale
increases, the market price decreases. Assume that the market price p and the number of
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units for sale x are related by the demand equation: p  3000  x . Assume that the cost
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C(x) of producing x items is governed by the equation C  x   2000  10 x and the revenue
R(x) generated by selling x units is governed by R  x   100 x .
5a. Write the cost as a function of price (hint start by solving the price equation for x, what
does this give you?).
5b. Write the revenue as a function of price.
5c. Write the profit as a function of price.
6. An oil spill makes a circular pattern around a ship such that the radius in feet grows as a
function of time in hours r  t   150 t . Find the area of the spill as a function of time.
7. A 20 ft by 10 ft rectangular pool has been built. If 50 cubic feet of water is pumped into the
pool per hour, write the water-level height (feet) as a function of time (hours).
8. A family is watching a fireworks display. If the family is 2 miles from where the fireworks
are being launched and the fireworks travel vertically, what is the distance between the
family and the fireworks as a function of the height above the ground.
9. A couple is about to sell their house. They bought the house for $172,000. When they sell
it they will pay a 6% commission. Write a function that represents the profit they will make
from the sale of their home as a function of the selling price.
10. A student works at Target making $7 per hour and the weekly number of hours worked
per week x varies. If Target withholds 25% of his earnings for taxes and Social Security,
write the function E(x) that expresses the student’s take-home pay each week. Find the
inverse function E-1(x). What does the inverse function tell you?
11a. I went shopping at Macy’s over the weekend. Everything in the store was discounted
20%. Purchases are subject to sales tax of 9.25%. Write a function to describe the amount
of money I will spend on your purchases as a function of the non-discounted price.
11b. Find the inverse of the function above. What does the inverse represent?
12. Your wage is $8.00 per hour plus $0.75 for each unit produced per hour. So your hourly
wage y in terms of the number of units produced is y  8  .75x. Find the inverse function.
What does each variable in the inverse function represent?