If $f(x)$ and $f^{-1}(x)$ are inverse functions of each other and $f(x)=2 x+5$, what is $f^{-1}(8)$ ?
We are given a function \(f(x) = 2x + 5\) and we are looking for the value of \(f^{-1}(8)\), where \(f^{-1}\) denotes the inverse function of \(f\). An inverse function essentially reverses the mapping of the original function. If \(f(x) = y\), then the inverse function satisfies \(f^{-1}(y) = x\). Consequently, to find \(f^{-1}(8)\), we need to find the value \(x\) such that \(f(x) = 8\).
To find \(x\) such that \(f(x) = 8\), we substitute the expression for \(f(x)\) into the equation:
\[2x + 5 = 8\]
Now, we solve for \(x\). First, subtract 5 from both sides:
\[2x = 8 - 5\]
\[2x = 3\]
Finally, divide by 2:
\[x = \frac{3}{2}\]
Thus, \(f^{-1}(8) = \frac{3}{2}\).
In real life, inverse functions are used to reverse processes. For example, if you know the temperature in degrees Fahrenheit (\(F\)) based on Celsius (\(C\)) as \(F = 1.8C + 32\), the inverse function allows you to calculate the temperature in Celsius if you already have the Fahrenheit reading. This is similar to how we solved for \(x\) given the output \(8\), effectively "reversing" the function \(f(x) = 2x + 5\).