Question

6a. here is a graph of g(x) = \sqrt3{x}. use the graph of g(x) = \sqrt3{x} to help you explain why there is only one real solution to every equation of the form \sqrt3{x} = a in which a is a real number.

Explanation:

To determine why the equation \(\sqrt[3]{x}=a\) (where \(a\) is a real number) has only one real solution, we analyze the graph of \(g(x)=\sqrt[3]{x}\):

1. Interpret the Equation as a Horizontal Line Intersection: The equation \(\sqrt[3]{x}=a\) is equivalent to finding the \(x\)-value(s) where \(g(x)=\sqrt[3]{x}\) intersects the horizontal line \(y = a\).
2. Analyze the Graph’s Behavior: The graph of \(g(x)=\sqrt[3]{x}\) is a continuous, strictly increasing curve (it rises from left to right without flattening or decreasing). For any horizontal line \(y = a\) (where \(a\) is real), a strictly increasing function will intersect a horizontal line at most once.
3. Conclusion on Solutions: Since \(g(x)=\sqrt[3]{x}\) is strictly increasing and continuous over all real numbers, every horizontal line \(y = a\) will intersect the graph of \(g(x)\) at exactly one point. This means there is exactly one \(x\)-value that satisfies \(\sqrt[3]{x}=a\) for any real \(a\).

Answer:

The graph of \(g(x)=\sqrt[3]{x}\) is strictly increasing (rises continuously from left to right) and continuous over all real numbers. For any horizontal line \(y = a\) (where \(a\) is real), a strictly increasing function intersects a horizontal line at exactly one point. Thus, \(\sqrt[3]{x}=a\) has exactly one real solution (the \(x\)-coordinate of the intersection) for any real \(a\).