Find the inverse of the function $f(x)=2 x-4$.
The problem asks us to find the inverse of the function \(f(x) = 2x - 4\). An inverse function \(g(x)\), often denoted as \(f^{-1}(x)\), essentially "reverses" the effect of the function \(f(x)\). Specifically, if \(f(a) = b\), then \(g(b) = a\). We are given four options to choose from, which suggests we need to derive the inverse and match it with the correct expression.
To find the inverse of a function \(f(x)\), we set \(y = f(x)\) and then solve for \(x\) in terms of \(y\). Starting with the given function:
\[y = 2x - 4\]
Our first goal is to isolate \(x\). Adding \(4\) to both sides of the equation gives:
\[y + 4 = 2x\]
Next, divide both sides by \(2\) to solve for \(x\):
\[x = \frac{y + 4}{2}\]
We can rewrite this expression as:
\[x = \frac{1}{2}y + \frac{4}{2} = \frac{1}{2}y + 2\]
Now that we have \(x\) expressed as a function of \(y\), we swap the roles of \(x\) and \(y\) to express the inverse function in the standard form \(g(x)\):
\[y = \frac{1}{2}x + 2\]
Therefore, the inverse function is \(g(x) = \frac{1}{2}x + 2\). Comparing this result with the provided options:
We see that our derived function matches option c.
In economics, an inverse function can represent the relationship between price and demand. If a company knows how demand changes with price, \(f(\text{price}) = \text{demand}\), they use the inverse function \(g(\text{demand}) = \text{price}\) to determine what price they should set to achieve a specific level of demand. Understanding how to calculate inverse functions allows analysts to flip these relationships easily to make informed business decisions.