Find the inverse function of the function $f(x)=5 x-3$.
To find the inverse of the function \(f(x) = 5x - 3\), we are looking for a function \(f^{-1}(x)\) that essentially "undoes" the operations applied by \(f(x)\). The original function \(f(x)\) takes an input \(x\), multiplies it by 5, and then subtracts 3. To reverse this, our inverse function should perform the inverse operations in reverse order: adding 3 to the input and then dividing the result by 5.
We begin by setting the function expression equal to a variable \(y\), representing the output:
\[y = 5x - 3\]
Our goal is to isolate \(x\) to express it as a function of \(y\). First, we add 3 to both sides of the equation:
\[y + 3 = 5x\]
Next, we divide both sides by 5 to solve for \(x\):
\[x = \frac{y + 3}{5}\]
This equation tells us that for any given output \(y\), the input \(x\) that produced it is \(\frac{y + 3}{5}\).
By convention, we represent the inverse function by switching the roles of \(x\) and \(y\). We replace \(y\) with \(x\) in our derived expression, yielding the inverse function:
\[f^{-1}(x) = \frac{x + 3}{5}\]
Comparing this result with the provided options, we can see that it matches the first option.
Finding the inverse of a function is like reversing a series of steps in a process. Imagine you have a currency exchange function that calculates the value in US dollars given an amount in another currency. If the exchange function is \(f(x) = 5x - 3\), where \(x\) is the input amount, the inverse function \(f^{-1}(x) = \frac{x + 3}{5}\) allows you to perform the "reverse exchange," converting the dollar amount back into the original currency. This skill is vital in programming, data processing, and any field where you need to undo operations to retrieve original data.