Problem 59 - College Algebra

Question

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=-\sqrt{16-x^{2}}$$

Step-by-Step Solution

Answer

Yes, the function has an inverse that is also a function. The function is one-to-one.

Step 1: Graph the given function

To understand the function, it is recommended to plot the function \( f(x) = -\sqrt{16 - x^{2}} \). The shape of the graph is half of a circle with a radius of 4 units below the x-axis. The domain of the function is \[-4,4\] and the range is \[-4,0\].

Step 2: Apply the Horizontal Line Test

To find out if a function has an inverse that is also a function, you can perform the horizontal line test. If every horizontal line intersects the graph at most once, then the original function is one-to-one, and thus, it will have an inverse that is also a function. When applying the horizontal line test to this graph, you'll notice that each horizontal line crosses the graph only once. Thus, the function is one-to-one.

Step 3: Conclusion

Based on the analysis of the graph and the horizontal line test performed, it can be concluded that the function \( f(x) = -\sqrt{16 - x^{2}} \) is one-to-one and has an inverse that is also a function.