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Chapter 2: Problem 80

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

Short Answer

Expert verified
The function \(f(x) = \operatorname{int}(x-2)\) does not have an inverse that is a function.

Step by step solution

01

Understanding the function

The function given is \(f(x) = \operatorname{int}(x-2)\). This is a floor function, which for any real number \(x\), outputs the greatest integer less than or equal to \(x\). In this case, it's \(x-2\), which means it will output the greatest integer less than or equal to \(x-2\).
02

Graphing the function

Using a graphing utility, plot the function \(f(x) = \operatorname{int}(x-2)\). The graph will look like a series of steps with each step being of unit length along the x-axis. The step will rise whenever \(x-2\) becomes an integer that is when \(x = 3, 4, 5, ...\) and so on.
03

Determining if the function has an inverse

Check if any horizontal line intersects the graph at more than one point. If so, then the function is not one-to-one and does not have an inverse that is a function. For this function, any horizontal line, say \(y = k\), will intersect the graph at an infinite number of points. Thus, the function is not one-to-one and hence does not have an inverse.

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