Problem 57 - 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). $$f(x)=|x-2|$$

Step-by-Step Solution

Answer

Yes, the function \(f(x)=|x-2|\) is one-to-one and therefore has an inverse that is also a function.

Step 1: Graph the function

Using a graphing utility, you should graph the function \(f(x)=|x-2|\). This function represents an absolute value, which always results in a positive output. So, the graph of this function will be a v-shaped graph with the vertex at (2,0).

Step 2: Determine whether the function is one-to-one

To determine if the function is one-to-one (which would mean its inverse is also a function), apply the horizontal line test. The horizontal line test states that if a horizontal line intersects a function's graph in more than one point, then the function is not one-to-one. As the graph will be a 'v' shape, any horizontal line that crosses the graph will intersect it in only one point. So, the function is one-to-one.