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Chapter 2: Problem 76

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). $$ f(x)=x^{2}-1 $$

Short Answer

Expert verified
The function \(f(x) = x^{2} - 1\) is not one-to-one, and does not have an inverse that is a function.

Step by step solution

01

Graph the function

First, create the graph for the function \(f(x) = x^{2} - 1\). This is a quadratic function which will create a parabola. The graph should show that the function crosses the y-axis at -1 (representing the -1 in the function equation) and crosses the x-axis at -1 and 1 (representing the solutions to the equation \(x^{2} - 1 = 0\), which are -1 and 1).
02

Apply the Horizontal Line Test

Apply the horizontal line test across the graph created in the previous step. If the graph intersects at more than one point when a horizontal line is drawn, this means that the function is not one-to-one and hence it does not have an inverse that is also a function.
03

Conclusion

From the horizontal line test, it's evident that horizontal lines intersect the graph at more than one point. Hence, it can be concluded that the given function is not one-to-one, and does not possess an inverse that is also a function.

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