Problem 80 - 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). $$f(x)=\operatorname{int}(x-2)$$

Step-by-Step Solution

Answer

Based on the graph of the function and the horizontal line test, it can be concluded that the function \(f(x) = \operatorname{int}(x - 2)\) is not a one-to-one function and does not have an inverse that is a function. This is due to the fact that for each value of \(f(x)\), there are multiple corresponding \(x\) values.

Step 1: Graphing the function

First, graph the function \(f(x) = \operatorname{int}(x - 2)\) using a graphing utility. This function maps every real number \(x\) into the largest integer less than or equal to \(x - 2\). Thus, for any \(x\) belonging to an interval \((n, n+1]\), where \(n\) is an integer, the value of \(f(x)\) is equal to \(n-2\). For example, when \(x\) is in the interval \((2,3]\), \(f(x) = 0\).

Step 2: Checking if the function is one-to-one

The function \(f(x)\) is one-to-one if every \(x\) corresponds to a unique \(f(x)\) and vice versa. By looking at the graph, observe that the function has a horizontal line at each integer value of \(y\) (or \(f(x)\)). To determine if the function is a one-to-one function, use the horizontal line test. If any horizontal line intersects the graph more than once, then the function is not one-to-one.

Step 3: Determining the inverseness of the function

By using the horizontal line test, it can be seen that multiple horizontal lines intersect the graph at more than one point. This indicates that for these values of \(f(x)\) there is more than one corresponding \(x\) value which means the function is not one-to-one and hence does not have an inverse that is a function.