Problem 42 - 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 \(f^{-1}\) $$f(x)=x^{2}-1, x \leq 0$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x)= \sqrt{x+1}\), \(x \geq -1\). For \(f(x)\), the domain is \((-\infty,0]\) and the range is \((-\infty,+\infty)\). For \(f^{-1}(x)\), the domain is \([-1,+\infty)\) and the range is \((-\infty,0]\).

Step 1: Find the Inverse Function

Start by replacing \(f(x)\) with \(y\), which gives the equation \(y=x^{2}-1\). Now, swap \(x\) and \(y\) so the equation becomes \(x=y^{2}-1\). To find the inverse, solve this equation for \(y\). You should get \(f^{-1}(x)= \sqrt{x+1}\), which represents the inverse function. Remember that since \(x \leq 0\) for \(f\), \(f^{-1}(x) \geq -1\).

Step 2: Graph the Functions

To graph these functions, understand that the graph of an inverse function is a reflection of the original function over the line \(y=x\). This means that each point \((a,b)\) on the graph of \(f\) corresponds to the point \((b,a)\) on the graph of \(f^{-1}\). Draw \(f(x)\) first and then reflect on the graph to plot \(f^{-1}(x)\).

Step 3: Find the Domain and Range

The domain of a function is the set of all possible inputs, and the range is the set of resultant outputs. For \(f(x)\), since \(x \leq 0\), the domain is \((-\infty,0]\). The range of \(f(x)\) is then \((-\infty,+\infty)\) because a squared term is always non-negative, and subtracting 1 allows it to span all real numbers. For the inverse function \(f^{-1}(x)\), since \(f^{-1}(x) \geq -1\), the domain is \([-1,+\infty)\) and the range is \((-\infty,0]\), which is just a reflection of the domain and range of the original function.