Problem 81 - 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). $$f(x)=|x-2|$$

Step-by-Step Solution

Answer

By graphing the function \(f(x) = |x-2|\), we see that the graph increases to the left and right of the point (2,0). Applying the Horizontal Line Test, we find that any horizontal line will touch the graph at more than one point. Therefore, the function is not one-to-one and does not have an inverse that is also a function.

Step 1: Graphing the Function

Construct the graph of the given function \(f(x) = |x-2|\). This is an absolute function which is shifted 2 units to the right. Hence, the graph should reflect this shift. The graph should increase to the left and right of the point (2,0).

Step 2: Applying the Horizontal Line Test

Now, apply the Horizontal Line Test to check if the function is one-to-one. This test is done by imagining or drawing a horizontal line on the graph, then checking if that line touches the graph at more than one point.

Step 3: Interpreting the Result

If the horizontal line touches the graph at only one point, the function passes the test and therefore, has an inverse that is also a function indicating it as one-to-one. However, if the line touches the graph at more than one point, then the function is not one-to-one and does not have an inverse that is a function.