Question

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 qunits 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 \(\mathrm{C}\) 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 \(C\). Illustrate this idea. Temperature Conversion The function \(F(C)=\frac{9}{5} C+32\) 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?

Short answer

The inverse function is \(C = \frac{5}{9}(F - 32)\). Verification: \(C(F(C)) = C\) and \(F(C(F)) = F\). At 70°F, the temperature is 21.11°C.

Step-by-step solution

Define the problem

Given the function that converts temperature from degrees Celsius to degrees Fahrenheit as \(F(C) = \frac{9}{5}C + 32\), the task is to find the inverse function \(C(F)\) that converts temperature back from degrees Fahrenheit to degrees Celsius.

Express Celsius as a function of Fahrenheit

To find \(C(F)\), solve the equation \(F(C) = \frac{9}{5}C + 32\) for \(C\):1. Start with \(F = \frac{9}{5}C + 32\).2. Subtract 32 from both sides to get \(F - 32 = \frac{9}{5}C\).3. Multiply both sides by \(\frac{5}{9}\) to solve for \(C\): \(C = \frac{5}{9}(F - 32)\).

Verify the inverse relationship

Show that \(C(F(C)) = C\) and \(F(C(F)) = F\):1. Calculate \(C(F(C))\):\(C(F(C)) = \frac{5}{9}(F(C) - 32) = \frac{5}{9}(\frac{9}{5}C + 32 - 32) = C\).2. Calculate \(F(C(F))\):\(F(C(F)) = \frac{9}{5}C(F) + 32 = \frac{9}{5}(\frac{5}{9}(F - 32)) + 32 = F\).

Calculate the Celsius temperature for 70°F

Using the function \(C = \frac{5}{9}(F - 32)\), substitute \(F = 70\) degrees:\(C = \frac{5}{9}(70 - 32) = \frac{5}{9} \times 38 = 21.11 \text{ degrees Celsius}\).

Key concepts

Inverse Functions

Understanding inverse functions is crucial especially in applications such as temperature conversion. An inverse function essentially reverses the operation of the original function. For example, if you have a function that converts Celsius to Fahrenheit, the inverse function would convert Fahrenheit back to Celsius.

To find an inverse function, you solve the original function equation for the independent variable. For instance, given the function that converts Celsius to Fahrenheit: \( F(C) = \frac{9}{5}C + 32 \) We solve for \(C\) to get:

1. Start with the equation: \( F = \frac{9}{5}C + 32 \).2. Isolate \(C\) by first subtracting 32 from both sides: \( F - 32 = \frac{9}{5}C \).3. Multiply both sides by \( \frac{5}{9} \) to solve for \(C\): \( C = \frac{5}{9}(F - 32) \).

This results in the inverse function, \( C = \frac{5}{9}(F - 32) \), which can be used to convert back from Fahrenheit to Celsius.

Function Notation

Function notation is a way to name functions for easy reference, using symbols like \(f(x)\), where \(f\) is the function and \(x\) is the input or independent variable. For example, the temperature conversion from Celsius to Fahrenheit can be written as \( F(C) = \frac{9}{5} C + 32 \). In applied problems, we might see different notations based on context.

For instance, in economics, you might see cost as a function of the number of units produced, such as \(C = C(q)\). Here, \(C\) represents cost, and \(q\) represents quantity. When finding inverses in applied problems, the notation might change to something like \(q = q(C)\) instead of the typical mathematical inverse notation \(f^{-1}(x)\). The critical thing is to understand the relationship expressed by the function, regardless of the notation used.

Temperature Scales

Temperature scales are systems of measuring temperature. The two most commonly used scales are Celsius (°C) and Fahrenheit (°F). Selecting the correct scale and converting between them is vital in many fields, such as science, engineering, and daily life.

To convert from Celsius to Fahrenheit, we use the formula: \( F(C) = \frac{9}{5}C + 32 \).
To convert from Fahrenheit to Celsius, we use the inverse formula derived earlier: \(C = \frac{5}{9}(F - 32)\).

For example, if you have a temperature of 70°F and need to convert it to Celsius:

1. Substitute 70 into the Fahrenheit to Celsius function: \( C = \frac{5}{9}(70 - 32) \).2. Perform the subtraction: \( 70 - 32 = 38 \).3. Multiply by \( \frac{5}{9} \): \( C = \frac{5}{9} \times 38 = 21.11 \text{°C} \).

This calculation shows that 70°F is equivalent to approximately 21.11°C. Understanding how to navigate these conversions helps in various practical and academic applications.