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Chapter 1: Problem 41

Use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. $$g(x)=(x+5)^{3}$$

Short Answer

Expert verified
Yes, the function \(g(x)=(x+5)^{3}\) has an inverse since the graph of the function passes the Horizontal Line Test.

Step by step solution

01

Graphing the function

To graph the function, plug \(x\) values into the function and plot the resulting \(y\) values. If a graphing utility is available, this will be an immediate step. Input the function \(g(x)=(x+5)^{3}\) into the graphing utility. This will provide a visual representation of the function.
02

Applying the Horizontal Line Test

The Horizontal Line Test involves checking if any horizontal line drawn through the graph of the function intersects the graph at more than one point. If it does, the function does not have an inverse. Inspect the graph of \(g(x)=(x+5)^{3}\). It's immediate from the shape of the graph that it's one-to-one, as any horizontal line will intersect our graph only once.
03

Determining invertibility

Since the graph of the function passes the Horizontal Line Test - every horizontal line intersects the graph at most once - the function \(g(x)=(x+5)^{3}\) has an inverse function.

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