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Which graph represents the function f(x)=sqrt [3](x-2) 7 B C D

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EloiseProfessional · Tutor for 6 years

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To determine which graph represents the function \(f(x) = \sqrt[3]{x-2}\), we need to consider the properties of the cubic root function and how the transformation affects its graph.Step 1: Understand the basic graph of \(f(x) = \sqrt[3]{x}\)The basic graph of \(f(x) = \sqrt[3]{x}\) is a monotonically increasing function that passes through the origin (0,0). As \(x\) approaches negative infinity, \(y\) approaches negative infinity, and as \(x\) approaches positive infinity, \(y\) approaches positive infinity.Step 2: Apply the transformationThe function \(f(x) = \sqrt[3]{x-2}\) is a horizontal shift of the basic cubic root function \(f(x) = \sqrt[3]{x}\) to the right by 2 units. This means that every point on the graph of \(f(x) = \sqrt[3]{x}\) is moved 2 units to the right to get the graph of \(f(x) = \sqrt[3]{x-2}\).Step 3: Identify key pointsThe point (0,0) on the basic graph of \(f(x) = \sqrt[3]{x}\) will be shifted to (2,0) on the graph of \(f(x) = \sqrt[3]{x-2}\). Additionally, the function will still be monotonically increasing, meaning it will continue to rise as \(x\) increases.Step 4: Consider the end behaviorAs \(x\) approaches negative infinity, the function \(f(x) = \sqrt[3]{x-2}\) will approach negative infinity, not 0 as described in the picture details. This is because the cubic root function does not have a horizontal asymptote; it continues to decrease without bound as \(x\) decreases.Based on the information provided in the picture details, there seems to be a discrepancy. The description of the graph in the picture details does not match the function \(f(x) = \sqrt[3]{x-2}\). The function \(f(x) = \sqrt[3]{x-2}\) does not have a horizontal asymptote as \(x\) approaches negative infinity, and it does not pass through (-2, -2) and (2, 2).Therefore, based on the correct properties of the function \(f(x) = \sqrt[3]{x-2}\), none of the options A, B, C, or D can be determined as the correct graph without seeing the actual images. The correct graph should show a monotonically increasing function that passes through the point (2,0) and continues to rise and fall without bound as \(x\) increases and decreases, respectively.

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Which graph represents the function f(x)=sqrt [3](x-2) 7 B C D

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