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Chapter 2: Problem 110

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=\sqrt[3]{x-2}$$

Short Answer

Expert verified
The graph of \(g(x)=\sqrt[3]{x-2}\) is a horizontal shift of the graph of the cube root function \(f(x)=\sqrt[3]{x}\), 2 units to the right.

Step by step solution

01

Graphing the Function \(f(x)=\sqrt[3]{x}\)

First, graph the basic function \(f(x)=\sqrt[3]{x}\). This function is a simple, untransformed cube root.
02

Understanding Transformations

A transformation is a change made to a graph which can result in its shift, reflection, stretch, or compression. In the function \(g(x)=\sqrt[3]{x-2}\), the '-2' inside the root causes a horizontal shift of the graph.
03

Graphing the Function \(g(x)=\sqrt[3]{x-2}\)

Based on our understanding of transformations, we know this graph will be a horizontal shift of the graph of \(f(x)=\sqrt[3]{x}\) 2 units to the right. Thus, every point \( P(x, y) \) on the graph of \(f(x)=\sqrt[3]{x}\) will correspond to a point \( P'(x+2, y) \) on the graph of \(g(x)=\sqrt[3]{x-2}\).

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