Question

Use a graphing utility to graph the function. Determine whether the function is one-to-one on its entire domain. $$ g(x)=(x+5)^{3} $$

Short answer

The function \(g(x)=(x+5)^3\) is one-to-one across its entire domain.

Step-by-step solution

Graph the function

Use a graphing utility to graph the function \(g(x)=(x+5)^3\). This function is a cubic function shifted five units to the left. The graph should show that each x-value corresponds to a unique y-value.

Analyze the graph

Look at the graph. Each horizontal line should cross the graph at most once to verify that it is a one-to-one function.

Conclusion

Since the graph confirms that each x-value has a unique y-value and each horizontal line crosses the graph once, we conclude the function \(g(x)=(x+5)^3\) is one-to-one across its entire domain.