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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
Based on the graph and the horizontal line test, the function \(f(x) = x^2 - 1\) does not have an inverse that is a function, because it is not one-to-one.

Step by step solution

01

Graph the Function

First, graph the function \(f(x) = x^2 - 1\) using a graphing utility. This function is a parabola shifted one unit down from the origin, opening upward.
02

Assess the Graph

Next, make an assessment of the generated graph. Deduce whether or not a horizontal line drawn through the graph crosses the plot more than once. For the function \(f(x) = x^2 - 1\), a horizontal line will intersect the graph at more than one point.
03

Conclusion

Based on the horizontal line test, the function \(f(x) = x^2 - 1\) is not one-to-one because any horizontal line above or equal to -1 intersects the graph at more than one point. Therefore, the function does not have an inverse that is also a function.

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