Which equation can be simplified to find the inverse of $y=x^2-7$?
To find the inverse of a function \(y = f(x)\), the fundamental step is to swap the roles of the independent variable \(x\) and the dependent variable \(y\). This reflects the input and output relationship, essentially reversing the function's operation. We are given the function \(y = x^2 - 7\), where \(x\) is the input and \(y\) is the output. To prepare this function for finding its inverse, we replace every instance of \(x\) with \(y\) and every instance of \(y\) with \(x\).
Starting with the original equation:
\[y = x^2 - 7\]
We now apply the swap by setting the variable \(x\) equal to the expression involving the new variable \(y\):
\[x = y^2 - 7\]
This equation is the direct result of swapping the variables \(x\) and \(y\) in the original function definition. It provides the foundation for isolating \(y\) to find the final expression for the inverse function \(y = f^{-1}(x)\).
We now compare the resulting equation \(x = y^2 - 7\) with the given options:
The equation derived from our swap operation, \(x = y^2 - 7\), corresponds exactly to option c. Thus, this is the correct equation to use for finding the inverse of the given function.
In real-life scenarios, finding an inverse function is essential for reversing processes. For example, if you have a formula that calculates the temperature in Fahrenheit (\(F\)) based on the temperature in Celsius (\(C\)), such as \(F = \frac{9}{5}C + 32\), finding the inverse allows you to calculate the temperature in Celsius if you already know the temperature in Fahrenheit. By swapping the variables and solving for the new dependent variable, you can transform the equation into \(C = \frac{5}{9}(F - 32)\), making it easy to perform the conversion in reverse. This demonstrates the practical utility of understanding how to manipulate equations to find inverses.