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Chapter 1: Problem 118

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 the function \(g(x) = \sqrt[3]{-x+2}\) is a reflection of the graph of the basic cube root function \(f(x) = \sqrt[3]{x}\) about the y-axis, and then shifted two units to the left.

Step by step solution

01

Understanding the Base Function

The cube root function \(f(x) = \sqrt[3]{x}\) is a function that is increasing, but at a decreasing rate as \(x\) increases. The graph crosses the x-axis and the y-axis at the origin (0, 0). It looks essentially like half of a parabola stood upright.
02

Identifying Transformations

In the given function \(g(x) = \sqrt[3]{-x+2}\), two transformations can be seen compared to the base function. The '-' sign before \(x\) in \(-x+2\) signifies a reflection about the y-axis. The '+2' signifies a shift of two units to the left.
03

Graphing the Transformed Function

Start with the basic graph of \(f(x) = \sqrt[3]{x}\). Reflect this graph in the y-axis, and then shift it to left by two units. This transformation will give you the graph of \(g(x) = \sqrt[3]{-x+2}\)

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