What is the inverse of the function $f(x)=2 x-10$ ?
To find the inverse function \(h(x) = f^{-1}(x)\) of the given function \(f(x) = 2x - 10\), we begin by setting the function equal to \(y\):
\[y = 2x - 10.\]
In the inverse function, the roles of \(x\) and \(y\) are reversed. Therefore, we solve for \(x\) in terms of \(y\) to understand how to map the output back to the input.
Starting from the equation \(y = 2x - 10\), we add \(10\) to both sides to isolate the term containing \(x\):
\[y + 10 = 2x.\]
Next, we divide both sides by \(2\) to isolate \(x\):
\[x = \frac{y + 10}{2}.\]
This can be simplified by distributing the division:
\[x = \frac{y}{2} + \frac{10}{2} = \frac{1}{2}y + 5.\]
Now that we have solved for \(x\) in terms of \(y\), we express the inverse function \(h(x)\) by replacing \(y\) with \(x\):
\[h(x) = \frac{1}{2}x + 5.\]
Comparing this result to the provided options, we see that it matches the option \(h(x) = \frac{1}{2}x + 5\).
Finding the inverse of a function is crucial in various real-life applications, such as currency conversion. For example, if a function converts Dollars to Euros with a specific exchange rate and fee, the inverse function allows you to quickly calculate how many Dollars you need to obtain a target amount of Euros. Similarly, in programming or data processing, inverse functions are used to reverse operations like encryption/decryption or data scaling, ensuring that systems can return to their original states after transformation.