Problem 39 - College Algebra Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=2 x-1$$

Step-by-Step Solution

Answer

The inverse of \(f(x) = 2x - 1\) is \(f^{-1}(x) = (x+1)/2\). The graphs of \(f\) and \(f^{-1}\) are reflections of each other over the line \(y = x\). The domain and range of both functions is all real numbers, represented in interval notation as (-∞, ∞).

Step 1: Find the Inverse Function

To find the inverse of \(f(x) = 2x - 1\), replace \(f(x)\) with \(y\). This gives \(y = 2x - 1\). Then swap \(x\) and \(y\) to get \(x = 2y - 1\). Solve this for \(y\) to get \(y = (x+1)/2\). This gives the inverse function, \(f^{-1}(x) = (x+1)/2\)

Step 2: Graph The Functions

To graph \(f(x) = 2x - 1\), plot the line with slope 2 and y-intercept -1. To graph \(f^{-1}(x) = (x+1)/2\), plot the line with slope 1/2 and y-intercept 1/2. The graphs should be reflections of each other over the line \(y = x\) because a function and its inverse are reflective of each other.

Step 3: Find The Domains and Ranges

For \(f(x) = 2x - 1\), the domain and range is all real numbers, since any \(x\) value can be input to get any \(y\) value. This is written in interval notation as (-∞, ∞). Since \(f^{-1}(x) = (x+1)/2\) is also a linear function, its domain and range is also all real numbers, so also (-∞, ∞).