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Chapter 2: Problem 116

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

Short Answer

Expert verified
The graph of \(h(x)\) will be a reflected version of \(f(x)\) (because of the negative sign) and shifted 2 units to the right (because of \(x-2\)). It decreases as \(x\) moves away from 2, and no longer passes through the origin.

Step by step solution

01

Graphing the Cube Root Function

Begin by graphing the parent function, which is the cube root function \(f(x) = \sqrt[3]{x}\). This function has a key point at (0,0), which means it passes through the origin. As \(x\) increases, \(\sqrt[3]{x}\) also increases. As \(x\) decreases, \(\sqrt[3]{x}\) also decreases. The graph, therefore, is a curve that increases from left to right, passing through the origin. The graph should be drawn appropriately.
02

Understanding the Transformations

Next, consider the function given, \(h(x) = -\sqrt[3]{x-2}\). This function can be seen as the function \(f(x)\) after two transformations. The first is a reflection over the x-axis (because of the negative sign), and the second is a shift 2 units to the right (because of \(x-2\)). The graph of \(h(x)\) will still be a curve, but this time it decreases as \(x\) moves away from 2, and it no longer passes through the origin.
03

Graphing the Transformed Function

Finally, the graph of \(h(x)\) should be drawn. The curve will look like the reflected version of \(f(x)\) and will be shifted 2 units to the right. The curves of \(f(x)\) and \(h(x)\) should be drawn on the same set of axes for a clearer comparison.

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