In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Temperature Conversion The function F ( C ) = 9 5 C + 32 converts a temperature from C degrees Celsius to F degrees Fahrenheit. (a) Express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F . (b) Verify that C = C ( F ) is the inverse of F = F ( C ) by showing that C ( F ( C ) ) = C and F ( C ( F ) ) = F . (c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Temperature Conversion The function F ( C ) = 9 5 C + 32 converts a temperature from C degrees Celsius to F degrees Fahrenheit. (a) Express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F . (b) Verify that C = C ( F ) is the inverse of F = F ( C ) by showing that C ( F ( C ) ) = C and F ( C ( F ) ) = F . (c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using
to represent a function, an applied problem might use
to represent the cost
of manufacturing q units of a good. Because of this, the inverse notation
used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as
will be
. So
is a function that represents the cost
as a function of the number
of units manufactured, and
is a function that represents the number
as a function of the cost
. Problems 91-94 illustrate this idea.
Temperature Conversion The function
converts a temperature from
degrees Celsius to
degrees Fahrenheit.
(a) Express the temperature in degrees Celsius
as a function of the temperature in degrees Fahrenheit
.
(b) Verify that
is the inverse of
by showing that
and
.
(c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?
Expert Solution
To determine
To find: The function converts a temperature from degrees Celsius to degrees Fahrenheit.
a. Express the temperature in degrees Celsius as a function of the temperature in degrees Fahrenheit .
Answer to Problem 92AYU
Solution:
a. The temperature in degrees Celsius as a function of the temperature in degrees Fahrenheit as .
Explanation of Solution
Given:
The function converts a temperature from degrees Celsius to degrees
Fahrenheit as .
Calculation:
a. We can express temperature in degrees Celsius as a function of the temperature in degrees Fahrenheit .
Given
Expert Solution
To determine
To find: The function converts a temperature from degrees Celsius to degrees Fahrenheit.
b. Verify that is the inverse of by showing that and .
Answer to Problem 92AYU
Solution:
b. Verified that is the inverse of .
Explanation of Solution
Given:
The function converts a temperature from degrees Celsius to degrees
Fahrenheit as .
Calculation:
b. Verify is the inverse of by showing below.
Similarly, we can prove
Expert Solution
To determine
To find: The function converts a temperature from degrees Celsius to degrees Fahrenheit.
c. What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?
Answer to Problem 92AYU
Solution:
c. The temperature in degrees Celsius if it is 70 degrees Fahrenheit as.
.
Explanation of Solution
Given:
The function converts a temperature from degrees Celsius to degrees Fahrenheit as .
Calculation:
c. The temperature in degrees Celsius if it is 70 degrees Fahrenheit as
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