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Chapter 1: Problem 44

Use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. $$h(x)=|x+4|-|x-4|$$

Short Answer

Expert verified
After graphing \( h(x) = |x+4| - |x-4| \) and applying the Horizontal Line Test, the determination of whether or not it has an inverse function can be made.

Step by step solution

01

Understanding the Function

Recognize \( h(x) = |x+4| - |x-4| \) as an absolute value function, which typically results in a 'V' shaped graph. Here, it is the difference between two absolute value functions, which will influence the shape of the graph.
02

Graphing the Absolute Value Function

Use a graphing utility to graph the function. Plot points for a variety of \( x \) values (consider using both negative, zero, and positive values). Remember that absolute value makes any negative input positive, so the graph will not dip below the \( x \)-axis.
03

Applying the Horizontal Line Test

Draw horizontal lines through various points of the graph. Observe whether each line intersects the graph at more than one point. If it does, then the function does not have an inverse. If no horizontal line intersects more than once, the function does have an inverse.

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