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Chapter 2: Problem 115

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt[3]{x+2}$$

Short Answer

Expert verified
The function \(h(x) = -\sqrt[3]{x + 2}\) is a transformation of \(f(x) = \sqrt[3]{x}\) that involves a reflection over the x-axis and a horizontal shift 2 units to the left.

Step by step solution

01

Start with the Base Function

Start by graphing the base function \(f(x)=\sqrt[3]{x}\). This is a function whose value for a given x is the cube root of x. It is asymmetric with respect to the origin. The graph starts at \(-\infty\) and rises with an increasing slope, crosses through the origin, and then slows down its growth as it extends toward \(+\infty\).
02

Identify the Transformation

Next, it is necessary to identify the transformations that will occur based on the function \(h(x) = -\sqrt[3]{x + 2}\). There are two transformations here: a reflection in the x-axis (indicated by the negative sign) and a horizontal move to the left by 2 units.
03

Apply the Reflection

The negative sign in \(h(x)\) indicates that the transformed function, when compared to the original function, will be reflected in the x-axis. The result will be a downward opening function.
04

Apply the Horizontal Shift

Now, apply the horizontal transformation. The \((x+2)\) in the cube root indicates a horizontal shift of 2 units to the left(which is opposite to the sign of 2). Adjust every point on the reflected graph by moving it two places to the left.
05

Graph the Final Function

Finally, draw out the final graph. This new graph represents \(h(x) = -\sqrt[3]{x + 2}\), a reflection of \(f(x)=\sqrt[3]{x}\) in the x-axis and shifted two units to the left.

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