Problem 69 - 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

Yes, \(f(x) = (9/5)*x + 32\) and \(g(x) = (5/9)*(x-32)\) are indeed inverse functions, as applying each one to the output of the other gives us the original input value. This feature is definitive of inverse functions.

Step 1: Substitute Function f into Function g

The first step is to substitute function \(f\) into function \(g\). That is, wherever there's an \(x\) in function \(g\), replace it with the entire expression of function \(f(x)\). So, the function \(g(f(x))\) will become \((5/9) * ((9/5)x + 32 - 32)\).

Step 2: Simplify the Resultant Function

The next step is to simplify this new function. The \(+32\) and \(-32\) terms would cancel each other out, while the constant terms in front of \(x\) would also cancel out, leaving us with \(x\) as the resultant function. Therefore, \(g(f(x)) = x\). This shows that if we apply \(f\) and then \(g\), we obtain the original value, fulfilling the main condition for \(f\) and \(g\) to be inverses.

Step 3: Substitute Function g into Function f

The final step is to substitute function \(g\) into function \(f\). Similarly, replace \(x\) in function \(f\) with the entire expression of function \(g(x)\). So, the function \(f(g(x))\) will become \((9/5)*((5/9)*(x-32)) + 32\).

Step 4: Simplify the Second Resultant Function

Upon simplifying, using the fact that \((9/5)*(5/9) = 1\), we get \(x\). Thus \(f(g(x)) = x\). Hence applying \(g\) and then \(f\), we also obtain the original value. This fulfills the other condition for \(f\) and \(g\) to be inverses.