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Chapter 2: Problem 83

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). $$ f(x)=-\sqrt{16-x^{2}} $$

Short Answer

Expert verified
Yes, the function \( f(x)=-\sqrt{16-x^{2}} \) is one-to-one, and thus has an inverse that is a function.

Step by step solution

01

Graph the function

Use a graphing utility to graph the function \( f(x)=-\sqrt{16-x^{2}} \). This will yield a semi-circle on the negative y-axis with a radius of 4 units.
02

Apply the horizontal line test

Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, the function is not one-to-one.
03

Draw conclusion

If no horizontal line intersects the graph more than once, the function is one-to-one, and it has an inverse that is also a function. In our case, no horizontal line intersects the graph of \( f(x) \) more than once, so the function is one-to-one.

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