SOLVING WORD PROBLEMS INVOLVING INVERSE FUNCTIONS

Problem 1 :

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 + 0.75x

a) Find the inverse function.

b) What does each variable represent in the inverse function?

c) Determine the number of units produced when your hourly wage is $22.25.

Solution:

y = 8 + 0.75x

a) To find the inverse function,

Interchange x and y,

x = 8 + 0.75y

x - 8 = 0.75y

b) In the inverse function, y represents the hourly wage, which was the dependent variable in the original equation, and x represents the number of units produced, which was the independent variable.

c)

y = 22.25

y = 8 + 0.75x

22.25 = 8 + 0.75x

14.25 = 0.75x

x = 19

Therefore, the number of units produced when the hourly wage is $22.25 is 19.

Problem 2 :

Diesel Mechanics The function given by

y = 0.03x² + 245.50, 0 < x < 100

approximates the exhaust temperature y in degrees Fahrenheit., where x is the percent load for a diesel engine.

a) Find the inverse function. What does each variable represent in the inverse function?

b) Use a graphing utility to graph the inverse function.

c) The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval?

Solution:

a)

y = 0.03x² + 245.50

To find inverse function

Interchange x and y.

x = 0.03y² + 245.50

0.03y² = x - 245.50

b) 

c) 0 < x < 92.11

Problem 3 :

You need a total of 50 pounds of two types of ground beef costing $1.25 and $1.60 per pound, respectively. A model for the total cost y of the two types of beef is

y = 1.25x + 1.60(50 - x)

where x is the number of pounds of the less expensive ground beef.

a) Find the inverse function of the cost function. What does each variable represent in the inverse function?

b) Use the context of the problem to determine the domain of the inverse function.

c) Determine the number of pounds of the less expensive ground beef purchased when the total cost is $73.

Solution:

a)

y = 1.25x + 1.60(50 - x)

To find inverse function

Interchange x and y.

x = 1.25y + 1.60(50 - y)

= 1.25y + 80.00 - 1.60y

= -0.35y + 80.00

b) 

x represents the total cost.

The total cost will be minimum if we buy 50 pounds of less expensive commodity.

Therefore, minimum value of x is 62.5

The total cost will be minimum if we buy 50 pounds of more expensive commodity.

Therefore, minimum value of x is 80

Domain of the inverse function is [62.50, 80].

c) 

Using the inverse function,

Number of pounds of less expensive commodity is y when x = 73

Problem 4 :

Miles Traveled The total numbers f (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time (in years) is given by t, with t = 5 corresponding to 1995.

a) Does f^-1 exist?

b) If f^-1 exists, what does it mean in the context of the problem?

c) If f^-1 exists, find f-1(2632).

d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year 2747 billion, would f^-1 exists? Explain.

Solution:

a) Yes

b) f^-1 yields the year for a given number of miles traveled by motor vehicles.

c) f^-1(2632) = 8

d) No. f(t) would not pass the Horizontal line test.