- Define the function as \(y = 9x + 3\), solve for \(x\) to find the inverse, and verify by checking composition: \(f^{-1}(x) = \frac{x-3}{9}\).
- Identify that for linear functions like these, both domain and range are all real numbers: \(D_f = R_f = D_{f^{-1}} = R_{f^{-1}} = (-\infty, \infty)\).
- Recognize that the graphs of \(f(x)\) and \(f^{-1}(x)\) are symmetric reflections of each other across the line \(y = x\).
- Final answers: \(f^{-1}(x) = \frac{x-3}{9}\); Domain and Range of \(f\) and \(f^{-1}\) are \((-\infty, \infty)\).
Explanation
- Analysis of the Function f(x) = 9x + 3
The function provided is \(f(x) = 9x + 3\). This is a linear function representing a straight line with a slope of \(9\) and a \(y\)-intercept of \(3\). Since the slope is non-zero, the function is one-to-one, meaning every \(y\)-value corresponds to exactly one \(x\)-value. This guarantees that an inverse function \(f^{-1}(x)\) exists. Our goal is to derive this inverse algebraically, determine the domain and range for both functions, and describe their graphical relationship.
- Finding and Verifying the Inverse Function
To find the inverse function \(f^{-1}(x)\), we set \(y = f(x)\), so \(y = 9x + 3\). We then solve for \(x\) in terms of \(y\):
Subtract 3 from both sides: \(y - 3 = 9x\).
Divide by 9: \(x = \frac{y - 3}{9}\).
Interchanging \(x\) and \(y\) to write the inverse function in terms of \(x\), we get \(f^{-1}(x) = \frac{x - 3}{9}\).
To verify this, we check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\):
\(f(f^{-1}(x)) = 9 \left(\frac{x - 3}{9}\right) + 3 = (x - 3) + 3 = x\).
\(f^{-1}(f(x)) = \frac{(9x + 3) - 3}{9} = \frac{9x}{9} = x\).
Both compositions equal \(x\), confirming that the inverse function is correctly identified as \(f^{-1}(x) = \frac{x - 3}{9}\).
- Domain and Range of f and f^{-1}
For the linear function \(f(x) = 9x + 3\), the domain consists of all possible input values, which for any linear polynomial is the set of all real numbers: \(D_f = (-\infty, \infty)\). Similarly, the range consists of all possible output values, and since the line extends infinitely in both directions, the range is also the set of all real numbers: \(R_f = (-\infty, \infty)\).
By definition, the domain of the inverse function \(f^{-1}\) is the range of the original function \(f\), and the range of \(f^{-1}\) is the domain of \(f\). Thus, for \(f^{-1}(x) = \frac{x - 3}{9}\):
Domain: \(D_{f^{-1}} = (-\infty, \infty)\).
Range: \(R_{f^{-1}} = (-\infty, \infty)\).
- Graphing the Functions and Understanding Symmetry
To graph these functions, we plot the line \(f(x) = 9x + 3\), which passes through points like \((0, 3)\) and \((-\frac{1}{3}, 0)\). The inverse function \(f^{-1}(x) = \frac{x - 3}{9}\) passes through points that are reflections of the points on \(f\), such as \((3, 0)\) and \((0, -\frac{1}{3})\). When plotted on the same Cartesian coordinate system, the graphs of \(f(x)\) and \(f^{-1}(x)\) are symmetric across the line \(y = x\). The line \(y = x\) acts as an axis of reflection, visually demonstrating that the two functions are inverse pairs.
Examples
In data encryption, functions are often used to scramble information (the function \(f\)) and then reverse that process to retrieve the original message (the inverse function \(f^{-1}\)). Just as \(f^{-1}(f(x)) = x\), an encryption algorithm transforms data into code, and the decryption key functions as the inverse, transforming the code back into the original, readable message. This process ensures secure communication in everyday digital activities like online banking or sending private messages.