Problem 83 - College Algebra Essentials

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

The function \( f(x)=-\sqrt{16-x^{2}} \) is not one-to-one, meaning it does not have an inverse that is a function.

Step 1: Graph the function

To graph the function \( f(x)=-\sqrt{16-x^{2}} \), plug in a series of x-values (both positive and negative), compute the corresponding y-values, and plot the points ((x, f(x)) on a graph. Additionally, draw the curve that best fits these points. A graphing utility will help with this step, plotting the function as a half circle center at (0,0). from [x=-4,x=4]. The vertical line test is passed which means this is indeed a function.

Step 2: Check if the function is one-to-one

Next, analyze the graph to determine if the function is one-to-one, or in other words, if it passes the horizontal line test. To do this, draw a horizontal line across the graph. If the line intersects the graph at more than one point, the function has failed the horizontal line test and is not one-to-one. In this case, the graph fails the horizontal line test because multiple x-values correspond to a single y-value. Thus, this function does not have an inverse that is a function.