Problem 45 - College Algebra

Question

The formula \(y=f(x)=\frac{9}{5} x+32\) is used to convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit. The formula $$y=g(x)=\frac{5}{9}(x-32)$$ is used to convert from \(x\) degrees Fahrenheit to \(y\) degrees Celsius. Show that \(f\) and \(g\) are inverse functions.

Step-by-Step Solution

Answer

Based on the results we have achieved, we can conclude that \(f(x)\) and \(g(x)\) are indeed inverse functions, as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Define the Functions

We have two functions; \[f(x) = \frac{9}{5} x + 32\] for the conversion from Celsius to Fahrenheit, and \[g(x) = \frac{5}{9} (x - 32)\] for the conversion from Fahrenheit to Celsius.

Step 2: Expressing One Function in Term of Other

We substitute \(f(x)\) into \(g(x)\) to see if we obtain \(x\) which is the primary way of proving inverse functions. Now, substitute \(f(x)\) as \(x\) in \(g(x)\), we get \[g(f(x)) = \frac{5}{9} (f(x) - 32) = \frac{5}{9}((\frac{9}{5}x+32)- 32)\]

Step 3: Simplify the Equation

Simplify this to get \(x\). After simplification, we obtain \[g(f(x)) = \frac{5}{9}(\frac{9x}{5}+32-32) = \frac{5}{9}\cdot\frac{9x}{5} = x\] The left-hand side and right-hand side are equal, which can be said as the output of function \(g(x)\) when \(x\) is \(f(x)\) is equivalent to \(x\). This is the condition for inverse functions.

Step 4: Verify Inverseness in the Other Way Around

We also substitute \(g(x)\) into \(f(x)\) to verify the inverseness the other way around: \[f(g(x)) = \frac{9}{5}g(x) + 32 = \frac{9}{5}(\frac{5}{9}(x-32)) + 32\] After simplification, we get: \[f(g(x)) = x\]

Key Concepts

Celsius to Fahrenheit Conversion

When you have a temperature measured in degrees Celsius and you want to convert it to degrees Fahrenheit, you use the function \( f(x) = \frac{9}{5}x + 32 \). This formula applies a linear transformation, which means it adjusts the scale and shifts the degrees by 32. Here’s how it works:

• Multiply the Celsius temperature by \( \frac{9}{5} \) (or 1.8). This step adjusts the scale of the temperature from Celsius to Fahrenheit, which means each degree Celsius maps to 1.8 degrees Fahrenheit.
• Add 32 to the result. This step accounts for the difference in zero points between Celsius, where 0 degrees is the freezing point of water, and Fahrenheit, where the freezing point is at 32 degrees.

This conversion is essential in many daily contexts, especially in countries that use the Fahrenheit system. You can easily check this by converting, say, 0 degrees Celsius and seeing that it results in 32 degrees Fahrenheit.

Fahrenheit to Celsius Conversion

To convert a temperature from Fahrenheit to Celsius, you use the function \( g(x) = \frac{5}{9}(x - 32) \). This function reverses the process used in Celsius to Fahrenheit conversion by performing the inverse operations:

• Subtract 32 from the Fahrenheit temperature. This shifts the temperature scale by removing the offset that was added during the initial conversion to Fahrenheit.
• Multiply the result by \( \frac{5}{9} \). This scales down the temperature back to the Celsius units, as each Fahrenheit degree expands to 0.555... (or \( \frac{5}{9} \)) of a degree Celsius.

This formula is invaluable for international travel and scientific work, ensuring the same temperature is consistently understood across metric and imperial systems.

Function Composition

Function composition involves applying one function to the results of another to see if the two functions reverse each other's effects. In our case, we explore this by substituting the output of one function into the other:- Start with \( f(x) \) which converts Celsius to Fahrenheit.- Next, apply \( g(x) \) to \( f(x) \). This process should convert the result back to Celsius if the two functions are truly inverses.In simpler terms, you input a Celsius temperature, convert it to Fahrenheit using \( f(x) \), and then back to Celsius with \( g(x) \). Doing so correctly returns you to your original temperature unit, confirming the integrity of the mathematical inverse.

Algebraic Proof of Inverses

To prove that two functions are inverses algebraically, you need to show that applying one function to the result of the other returns the original input value, that is, both \( g(f(x)) = x \) and \( f(g(x)) = x \).Start with \( g(f(x)) \):- Substitute \( f(x) = \frac{9}{5}x + 32 \) into \( g(x) = \frac{5}{9}(x - 32) \).- Simplify: \[ g(f(x)) = \frac{5}{9}((\frac{9}{5}x + 32) - 32) \]- Continue simplifying to confirm \( g(f(x)) = x \).Check the other way with \( f(g(x)) \):- Substitute \( g(x) \) back into \( f(x) \).- Simplify: \[ f(g(x)) = \frac{9}{5}(\frac{5}{9}(x - 32)) + 32 \]- This also results in \( f(g(x)) = x \).When both compositions yield the identity function, it confirms that \( f \) and \( g \) are indeed inverse functions.