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Chapter 2: Problem 117

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
Graph the original function \(f(x)=\sqrt[3]{x}\) , reflect it over the y-axis due to the 'negative x' term and then shift the resulting graph 2 units to the right due to the '-2 term' in the function \(g(x)=\sqrt[3]{-x-2}\) .

Step by step solution

01

Graph the original function

First, graph the function \(f(x)=\sqrt[3]{x}\). Draw x and y axes on your graph and plot points for x = -1, 0, and 1. The cube root of these values will provide the corresponding y values. Connect the points to get the original graph.
02

Reflection over the y-axis

The negative sign before x in the function \(g(x)=\sqrt[3]{-x-2}\) implies a reflection over the y-axis. Reflect the graph of \(f(x) \) over the y-axis to form a mirror image.
03

Horizontal Shift

The -2 in the function indicates a horizontal shift to the right by 2 units. Shift the reflected graph 2 units to the right to form the required graph of \(g(x)=\sqrt[3]{-x-2}\) .
04

Label the Function

Finally, indicate your function \(g(x)=\sqrt[3]{-x-2}\) on the graph clearly. Make sure the key points and the shape of the function are visible.

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