Problem 118
Question
Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=\sqrt[3]{-x+2}$$
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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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