Problem 69 - College Algebra Essentials

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

Indeed, the functions \(f\) and \(g\) are inverses of each other, as applying one and then the other results in the same original value, \(x\).

Step 1: Apply the composition \(f(g(x))\)

We substitute \(g(x)\) into \(f(x)\) i.e. we evaluate \(f(g(x))\). So it will be \(f\left(\frac{5}{9}(x-32)\right)\).

Step 2: Simplify the expression

By substituting \(g(x)\) into \(f(x)\), we get \(f\left(\frac{5}{9}(x-32)\right) = \frac{9}{5}\left(\frac{5}{9}(x-32)\right) + 32\). The \(\frac{9}{5}\) and \(\frac{5}{9}\) cancel out and we are left with \(x-32+32\), which simplifies to \(x\).

Step 3: Apply the composition \(g(f(x))\)

We substitute \(f(x)\) into \(g(x)\) i.e. we evaluate \(g(f(x))\). So it will be \(g\left(\frac{9}{5}\cdot x + 32\right)\).

Step 4: Simplify the Expression

By substituting \(f(x)\) into \(g(x)\) we get : \(g\left(\frac{9}{5}\cdot x + 32\right) = \frac{5}{9}\left(\left(\frac{9}{5}\cdot x + 32\right) - 32 \right)\). The \(\frac{5}{9}\) and \(\frac{9}{5}\) cancel out and we are left with \(x + 32 - 32\), which simplifies to \(x\).

Step 5: Draw Conclusion

Since \(f(g(x))=x\) and \(g(f(x))=x\), it's confirmed that functions \(f\) and \(g\) are inverses of each other.

Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another. In this exercise, you see function composition play out when you evaluate two functions to determine if they are inverses. Essentially, you are plugging the output from one function into the input of another. Here's how it works:
- When you have two functions \(f\) and \(g\), composing them results in \(f(g(x))\) or \(g(f(x))\).
- To check if two functions are inverses, compute \(f(g(x))\) and \(g(f(x))\). If both simplify to \(x\), then \(f\) and \(g\) are inverse functions.
This is exactly what happens in our exercise with temperature conversions, showing the relationship between Celsius and Fahrenheit through inverse functions by composition.

Algebraic Simplification

Algebraic simplification involves refining expressions to their simplest form. It helps in understanding and solving problems quickly with minimal errors. In function composition, this becomes especially important.

For instance, when we combine the functions like in \(f(g(x))\), the break down is:

• The expression \(f\left(\frac{5}{9}(x-32)\right)\) becomes \(\frac{9}{5}\left(\frac{5}{9}(x-32)\right) + 32\).
• The fractions \(\frac{9}{5}\) and \(\frac{5}{9}\) cancel each other out.
• This leaves us with \(x-32+32\), which simplifies to \(x\).

This simplification process is crucial in establishing that two functions are inverses, as seen in our Celsius-Fahrenheit conversion formulas.

Celsius to Fahrenheit Conversion

The conversion from Celsius to Fahrenheit relies on a straightforward formula. This helps in transforming temperatures from one scale to another, a necessity in many scientific and daily applications.

The formula is given by:\[F = \frac{9}{5} C + 32\]Where:

• \(F\) is the temperature in degrees Fahrenheit.
• \(C\) is the temperature in degrees Celsius.

This equation increases the Celsius temperature by the factor of \(\frac{9}{5}\), and then adds 32 to account for the zero shift between the two scales. This mathematical expression allows for seamless temperature conversions.

Fahrenheit to Celsius Conversion

To convert temperatures from Fahrenheit to Celsius, a reverse conversion formula is applied, reflecting the inverse relationship between the scales.

The formula is:\[C = \frac{5}{9}(F - 32)\]Where:

• \(C\) stands for temperature in degrees Celsius.
• \(F\) represents temperature in degrees Fahrenheit.

Here, you subtract 32 from the Fahrenheit temperature and then scale down by the factor \(\frac{5}{9}\), revealing how these scales are mathematically linked. This inverse approach complements the Celsius to Fahrenheit conversion, highlighting their mutual dependency and how one function undoes the work of the other.