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Chapter 2: Problem 108

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 obtained by taking the graph of the cube root function \(f(x) = \sqrt[3]{x}\) and translating it two units downwards.

Step by step solution

01

Graphing the parent function (Cube Root Function)

Begin by drawing the graph of the cube root function \(f(x) = \sqrt[3]{x}\). This function looks like a sideways parabola. For all \(x > 0\), the function increases as \(x\) increases. For \(x < 0\), the function decreases as \(x\) decreases.
02

Understanding the transformation

The given function \(g(x) = \sqrt[3]{x} - 2\) is derived from \(f(x) = \sqrt[3]{x}\) by applying a vertical shift (translation) of 2 units downwards. Basically, every point on the graph of \(f(x) = \sqrt[3]{x}\) is moved 2 units downwards to get the graph of \(g(x) = \sqrt[3]{x} - 2\).
03

Graphing the transformed function

Draw the graph of the transformed function \(g(x) = \sqrt[3]{x} - 2\) by shifting each point on the graph of \(f(x) = \sqrt[3]{x}\) by 2 units downwards. For example, the point (1, 1) on the graph of \(f(x)\) becomes the point (1, -1) on the graph of \(g(x)\).

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