Find the inverse of each of the following functions and sketch their graphs:
A linear function is typically expressed in the form \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Finding the inverse of a function, denoted as \(f^{-1}(x)\), involves essentially reversing the operations performed by the original function. Geometrically, the inverse function is a reflection of the original function across the identity line \(y = x\). To find the inverse, we replace \(f(x)\) with \(y\), swap the variables \(x\) and \(y\), and solve the resulting equation for \(y\).
For the function \(f(x) = x - 2\), we start by setting \(y = x - 2\). To find the inverse, we swap \(x\) and \(y\), yielding \(x = y - 2\). Solving for \(y\), we add \(2\) to both sides, which results in \(y = x + 2\). Thus, the inverse function is \(f^{-1}(x) = x + 2\).
For the function \(g(x) = 5x - 6\), we set \(y = 5x - 6\). Swapping the variables gives \(x = 5y - 6\). To isolate \(y\), we first add \(6\) to both sides, getting \(x + 6 = 5y\), and then divide by \(5\). This gives \(y = \frac{x + 6}{5}\), or equivalently \(g^{-1}(x) = \frac{1}{5}x + \frac{6}{5}\).
To sketch these graphs, we plot each function as a line. For \(f(x) = x - 2\), the line passes through \((0, -2)\) and \((2, 0)\). Its inverse \(f^{-1}(x) = x + 2\) passes through \((0, 2)\) and \((-2, 0)\). For \(g(x) = 5x - 6\), the line passes through \((0, -6)\) and \((1, -1)\). Its inverse \(g^{-1}(x) = \frac{x+6}{5}\) passes through \((0, 1.2)\) and \((-6, 0)\). All these lines reflect across the line \(y = x\). If a point \((a, b)\) lies on the original function, then the point \((b, a)\) lies on its inverse. Plotting these confirms the reflection property across the line \(y = x\).
Inverse functions are essential in fields where you need to reverse a process. For instance, in currency conversion, if a function \(f(x) = 1.1x\) converts Euros to Dollars, its inverse \(f^{-1}(x) = \frac{x}{1.1}\) allows you to easily convert back from Dollars to Euros. By mastering how to find and graph these inverses, you gain the ability to navigate bidirectional systems in finance, programming, and data analysis efficiently.