- The inverse of \(f(x) = 7x\) is found by swapping variables to solve for \(y\), resulting in \(f^{-1}(x) = \frac{x}{7}\).
- Both \(f(x)\) and \(f^{-1}(x)\) have a domain and range of \((-\infty, \infty)\).
- The functions \(f(x)\) and \(f^{-1}(x)\) are reflections of each other across the line \(y = x\).
- Final answers: \(f^{-1}(x) = \frac{x}{7}\); Domain and Range of \(f\) and \(f^{-1}\) are all real numbers. \(\boxed{f^{-1}(x) = \frac{x}{7}, \text{Domain: } (-\infty, \infty), \text{Range: } (-\infty, \infty)}\)
Explanation
- Finding and Verifying the Inverse Function
The function \(f(x) = 7x\) is a linear function. A one-to-one function has an inverse function denoted as \(f^{-1}(x)\). By definition, \(f^{-1}(x)\) is the function such that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). To find the inverse, we set \(y = f(x)\), so \(y = 7x\). Solving for \(x\) in terms of \(y\), we get \(x = \frac{y}{7}\). Swapping the variables \(x\) and \(y\) to express the inverse function, we obtain \(f^{-1}(x) = \frac{x}{7}\).
To check the answer:
- \(f(f^{-1}(x)) = f\left(\frac{x}{7}\right) = 7 \left(\frac{x}{7}\right) = x\).
- \(f^{-1}(f(x)) = f^{-1}(7x) = \frac{7x}{7} = x\).
Both compositions yield \(x\), confirming that \(f^{-1}(x) = \frac{x}{7}\) is indeed the inverse of \(f(x) = 7x\).
- Determining Domain and Range
The function \(f(x) = 7x\) is defined for all real numbers. Thus, the domain of \(f\) is \((-\infty, \infty)\). Since \(7x\) covers all real values as \(x\) varies through all real numbers, the range of \(f\) is \((-\infty, \infty)\).
For the inverse function \(f^{-1}(x) = \frac{x}{7}\), its domain is also the set of all real numbers, \((-\infty, \infty)\). Similarly, its range is the set of all real numbers, \((-\infty, \infty)\).
Generally, the domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\). Here, both are \((-\infty, \infty)\).
- Graphing the Functions and the Line y = x
The graphs of \(f(x) = 7x\) and \(f^{-1}(x) = \frac{x}{7}\) are straight lines passing through the origin \((0, 0)\).
- \(f(x) = 7x\) is a line with a slope of \(7\).
- \(f^{-1}(x) = \frac{x}{7}\) is a line with a slope of \(\frac{1}{7}\).
- The identity function \(y = x\) is a line with a slope of \(1\) passing through the origin at an angle of \(45^\circ\).
These lines exhibit symmetry about the line \(y = x\), which is a characteristic property of a function and its inverse. Reflection of \(f(x) = 7x\) across the line \(y=x\) results in \(f^{-1}(x) = \frac{x}{7}\).
Examples
Inverse functions are essential in fields like finance and engineering. For example, if you know a function \(f(x)\) that converts a certain amount of currency (like USD to EUR) based on an exchange rate of 7, then the inverse function \(f^{-1}(x)\) allows you to perform the reverse calculation (converting EUR back to USD). Understanding inverse functions helps in reversing operations, solving equations, and modeling relationships where one variable depends on another.