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Chapter 2: Problem 112

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2} \sqrt[3]{x-2}$$

Short Answer

Expert verified
The graph of \(h(x)\) is a cube root function with half the vertical scale and shifted 2 units to the right from the graph of \(f(x)\).

Step by step solution

01

Graph the function \(f(x)=\sqrt[3]{x}\)

This function produces a curve with a shape that increases slowly for negative x, goes through the origin, and then increases more steeply for positive x. The curve is symmetric about the origin.
02

Apply the vertical scaling.

The function \(h(x)\) takes the cube root function and scales it vertically by a factor of 1/2, this operation is represented by the equation \(h(x) = 1/2 f(x)\). Taking each vertical coordinate in the original graph and multiplying it by 1/2. This squishes the graph toward the x-axis.
03

Apply the horizontal shifting

Lastly, the function \(h(x)\) replaces \(x\) with \(x - 2\), which has the effect of shifting the graph 2 units to the right. This operation is represented by the equation \(h(x) = f(x - 2)\), which means replace every occurrence of \(x\) with \(x - 2\) in the original function \(f(x)\). It results in each point on the graph of \(f(x)\) being moved 2 units to the right to produce the graph of \(h(x)\).

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