Problem 56 - 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). $$f(x)=\operatorname{int}(x-2)$$

Step-by-Step Solution

Answer

Checking the graph of \(f(x) = \operatorname{int}(x-2)\) against the horizontal line test, it appears that the function isn't one-to-one. Therefore, it does not have an inverse that is also a function.

Step 1: Understanding the function

The function given is \(f(x)=\operatorname{int}(x-2)\). This represents the greatest integer less than or equal to \(x-2\). Known as the floor function or integer part function, it is a step function that takes a real number and gives the largest integer that is less than or equal to the number.

Step 2: Graphing the function

Utilizing a graphing utility, the function \(f(x) = \operatorname{int}(x-2)\) can be plotted. The x-axis would contain the input values, or 'x' values, while the y-axis contains the output, or 'f(x)'. Take a handful of whole values of 'x', plug them into the function to get the corresponding 'f(x)', plot these points and connect them to form the graph

Step 3: Determining the inverse

Once the graph has been plotted, it needs to be determined if the function has an inverse that is also a function. This means that the function must be one-to-one. A good test for determining this is the horizontal line test. If any horizontal line drawn through the function's graph intersects the graph in more than one place, the function is not one-to-one and does not have an inverse that is also a function.