Problem 85 - Algebra and Trigonometry

Question

Temperature Scales The relationship between the Fahrenheit \((F)\) and Celsius \((C)\) scales is given by $$ F(C)=\frac{9}{5} C+32 $$ (a) Find \(F^{-1} .\) What does \(F^{-1}\) represent? (b) Find \(F^{-1}(86) .\) What does your answer represent?

Step-by-Step Solution

Answer

(a) \(F^{-1}(x) = \frac{5}{9}(x - 32)\); it converts Fahrenheit to Celsius. (b) \(F^{-1}(86) = 30\); 86°F is 30°C.

Step 1: Understanding the Problem

We are given a formula to convert Celsius temperature (\(C\)) to Fahrenheit (\(F\)): \(F(C) = \frac{9}{5} C + 32\). We need to find the inverse function \(F^{-1}(C)\), which will give us the Celsius temperature when given a Fahrenheit temperature. Then, we need to find \(F^{-1}(86)\) and interpret its meaning.

Step 2: Solving for the Inverse Function

To find the inverse function \(F^{-1}(C)\), we start by setting \(F(C) = x\), where \(x\) is the Fahrenheit temperature. The equation becomes \(x = \frac{9}{5} C + 32\). We solve for \(C\) in terms of \(x\).1. Subtract 32 from both sides: \(x - 32 = \frac{9}{5} C\).2. Multiply both sides by \(\frac{5}{9}\) to isolate \(C\): \[C = \frac{5}{9}(x - 32)\].Thus, the inverse function is \(F^{-1}(x) = \frac{5}{9}(x - 32)\). \(F^{-1}(x)\) represents converting Fahrenheit to Celsius.

Step 3: Finding the Inverse at a Specific Point

To find \(F^{-1}(86)\), we substitute 86 for \(x\) in the inverse function:1. Use the formula \(F^{-1}(x) = \frac{5}{9}(x - 32)\).2. Substituting \(x = 86\): \[C = \frac{5}{9}(86 - 32)\].3. Calculate \(86 - 32 = 54\).4. Then, \(C = \frac{5}{9} \times 54 = 30\).Therefore, \(F^{-1}(86) = 30\), which represents 86 degrees Fahrenheit is equivalent to 30 degrees Celsius.

Key Concepts

Fahrenheit to Celsius

Temperature conversion between Fahrenheit and Celsius is essential for understanding and comparing temperatures across different scales. In the equation given, the conversion from Celsius (C) to Fahrenheit (F) is represented as:\[ F(C) = \frac{9}{5} C + 32 \]This formula is used to turn a temperature measured in Celsius into its Fahrenheit equivalent. The first component, \(\frac{9}{5} C\), scales the Celsius value, while 32 is an adjustment factor that aligns with the Fahrenheit scale. This adjustment accounts for the differing starting points of the two scales, with 0°C equating to 32°F.

• The factor \(\frac{9}{5}\) reflects the ratio between each degree increment in Celsius and Fahrenheit.
• The addition of 32 converts Celsius to the appropriate Fahrenheit measure.

Understanding this conversion is crucial when measuring temperatures in scientific contexts or when traveling to regions using different temperature units.

Inverse Functions

Inverse functions allow us to reverse operations—from converting Celsius to Fahrenheit, we shift to converting Fahrenheit back to Celsius.By finding the inverse function \(F^{-1}(x) = \frac{5}{9}(x - 32)\),we reveal how to translate temperatures in the opposite direction—from Fahrenheit to Celsius.

• The goal is to isolate the input variable, in this case, Celsius, from the equation. By solving for \(C\): 1. Set \(F(C) = x\), where \(x\) represents a Fahrenheit temperature. This becomes \(x = \frac{9}{5} C + 32\). 2. Subtract 32 to move constants to the Fahrenheit side: \(x - 32 = \frac{9}{5} C\). 3. Multiply by \(\frac{5}{9}\) to solve for \(C\): \(C = \frac{5}{9}(x - 32)\).
• This inverse function is essential for anyone needing to convert temperature readings from Fahrenheit to Celsius accurately.

It highlights the operational reversal where adding 32 becomes subtracting 32, and multiplying by \(\frac{9}{5}\) becomes multiplying by \(\frac{5}{9}\).

Linear Equations

The conversion between Fahrenheit and Celsius is an example of a linear relationship. Linear equations like\[ y = mx + b \]model straight-line relationships between two variables. In our temperature conversion:- The output (Fahrenheit, F) is derived from input (Celsius, C) using the slope \(\frac{9}{5}\), which signifies how changes in temperature correlate between the two scales.- The y-intercept, 32, indicates where the line crosses the y-axis, representing the baseline Fahrenheit equivalent of 0 degrees Celsius.Understanding linear equations helps in:

• Recognizing the constant rate of change indicated by the slope \(\frac{9}{5}\).
• Analyzing how incremental shifts in one temperature scale affect the other through the equation.

This foundational knowledge inherently applies not only to temperature conversion but also to any linear transformations where consistent proportional changes are present.