Problem 52 - 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). $$f(x)=x^{2}-1$$

Step-by-Step Solution

Answer

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

Step 1: Plot the function

The function \(f(x)=x^{2}-1\) is a quadratic function whose graph is a downward shifted parabola. By putting different values of x, obtain the y-values and plot these points on the graph to get a parabola shape graph.

Step 2: Check One-to-One

Once the graph has been drawn, proceed by using the horizontal line test to check if it is a one-to-one function. Place a horizontal line on the graph. If any line passes through more than one point on the graph, the function is not one-to-one.

Step 3: Conclude

For the function \(f(x)=x^{2}-1\), the parabola shape graph intersects any horizontal line more than once, hence, the function is not one-to-one, and therefore does not have an inverse that is a function.