Problem 40 - Algebra 2

Question

For Exercises 39 and \(40,\) use the following information. A formula for converting degrees Celsius to Fahrenheit is \(F(x)=\frac{9}{5} x+32\). Explain what purpose \(F^{-1}(x)\) serves.

Step-by-Step Solution

Answer

The inverse function \( F^{-1}(x) \) converts Fahrenheit temperatures back to Celsius.

Step 1: Understand the function

The function \( F(x) = \frac{9}{5}x + 32 \) is used to convert a temperature \( x \) in degrees Celsius to degrees Fahrenheit. For example, if \( x = 0 \), Celsius, then \( F(0) = \frac{9}{5}(0) + 32 = 32 \) degrees Fahrenheit.

Step 2: Purpose of the Inverse Function

The purpose of an inverse function \( F^{-1}(x) \) is to reverse the transformation that the original function \( F(x) \) performs. In other words, \( F^{-1}(x) \) converts a given temperature \( x \) from degrees Fahrenheit back to degrees Celsius.

Step 3: Derive the Inverse Function

To find \( F^{-1}(x) \), we need to solve the equation \( y = \frac{9}{5}x + 32 \) for \( x \). Start by setting \( y = \frac{9}{5}x + 32 \), then isolate \( x \). This involves subtracting 32 from both sides to get \( y - 32 = \frac{9}{5}x \), and then multiplying both sides by \( \frac{5}{9} \) to solve for \( x \). This results in \( x = \frac{5}{9}(y - 32) \).

Step 4: Express the Inverse Function

Now substituting \( y \) with \( x \) in the expression we derived, we have the inverse function \( F^{-1}(x) = \frac{5}{9}(x - 32) \). This function converts degrees Fahrenheit to degrees Celsius.

Key Concepts

Degrees Celsius

The Celsius scale, also known as the centigrade scale, is a temperature scale used to measure how hot or cold something is. It's widely used around the world, particularly in scientific contexts. The scale is named after the Swedish astronomer Anders Celsius, who developed it in 1742. It is based on the freezing and boiling points of water.

• Water freezes at 0 degrees Celsius.
• Water boils at 100 degrees Celsius.

This scale makes it easy to understand and convert comparisons between various temperature levels. Celsius is part of the metric system, which is used globally and is based on the number 10 for simplicity. When converting Celsius to other temperature scales, knowing the formulae and relationships is essential, as seen in converting Celsius to Fahrenheit.

Degrees Fahrenheit

The Fahrenheit scale is another way to measure temperature. It was developed by Daniel Gabriel Fahrenheit, a physicist and engineer, in the early 18th century. This scale is still commonly used in the United States.

• Water freezes at 32 degrees Fahrenheit.
• Water boils at 212 degrees Fahrenheit.

These are the defining reference points of this temperature scale. Although not as intuitive as the Celsius scale due to its seemingly arbitrary fix points, Fahrenheit allows for finer temperature increment divisions due to its higher granularity between freezing and boiling points. In the context of converting between the scales, the formula to convert Celsius to Fahrenheit is given by: \[ F(x) = \frac{9}{5}x + 32 \]This transformation scales the Celsius value and shifts it accordingly to match the Fahrenheit scale.

Function Transformation

Function transformation is a mathematical process that modifies a function to change its shape or position on a graph. In the context of temperature conversion, it modifies how we express degrees for different scales.The given formula \( F(x) = \frac{9}{5}x + 32 \) is a linear transformation, converting Celsius to Fahrenheit:

• The term \( \frac{9}{5}x \) scales the Celsius temperature.
• The +32 shifts it to align with the Fahrenheit scale's starting point.

The concept of an inverse function is crucial here. It effectively reverses the transformation. For the function \( F(x) = \frac{9}{5}x + 32 \), the inverse \( F^{-1}(x) = \frac{5}{9}(x - 32) \) converts Fahrenheit back to Celsius by reversing each step of the original transformation.

• First, it removes the 32 shift by subtracting.
• Then, it reverses the scaling by multiplying by \( \frac{5}{9} \).

Such transformations and their reversals are foundational in mathematics, providing insight into relationships between various forms of data, such as temperature.