Question

use the horizontal - line test to determine whether each function f has an inverse function. if so, draw a rough sketch of ( f^{-1} ) by reflecting f across ( y = x ). 8. (with a graph of a function f)

Explanation:

Step1: Apply Horizontal - Line Test

The horizontal - line test states that a function \(y = f(x)\) has an inverse function if and only if no horizontal line intersects the graph of \(f\) more than once. For the given graph of the function \(f\), we observe the graph. The graph of \(f\) is a curve that is strictly increasing (as \(x\) increases, \(y\) increases) and passes the horizontal - line test because any horizontal line will intersect the graph at most once. So, the function \(f\) is one - to - one and thus has an inverse function \(f^{-1}\).

Step2: Reflect the Graph across \(y = x\)

To sketch the graph of \(f^{-1}\), we use the property that the graph of an inverse function \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y=x\). For any point \((a,b)\) on the graph of \(f\), the point \((b,a)\) will be on the graph of \(f^{-1}\). So, we take key points on the graph of \(f\) (for example, if the graph of \(f\) passes through the origin \((0,0)\), which it seems to do from the sketch, then \((0,0)\) will also be on \(f^{-1}\) since reflecting \((0,0)\) across \(y = x\) gives \((0,0)\)). We then reflect the entire curve of \(f\) across the line \(y=x\) to get the rough sketch of \(f^{-1}\).

Answer:

The function \(f\) passes the horizontal - line test (is one - to - one), so it has an inverse function. To sketch \(f^{-1}\), reflect the graph of \(f\) across the line \(y = x\). The rough sketch of \(f^{-1}\) will be the mirror image of \(f\)'s graph with respect to the line \(y=x\).