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Chapter 2: Problem 118

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 function \(g(x)=\sqrt[3]{-x+2}\) is the result of reflecting the graph of \(f(x)=\sqrt[3]{x}\) in the y-axis and then shifting it 2 units to the right.

Step by step solution

01

Graph the Original Function

Begin by graphing the original function \(f(x)=\sqrt[3]{x}\). This is a cubic root function which roughly resembles a sideways S-shape. Both the x-axis and y-axis are symmetrical.
02

Identify the Transformations

In the function \(g(x)=\sqrt[3]{-x+2}\), two transformations are at play. Firstly, there is a reflection about the y-axis given by the negative sign before the x. Secondly, a horizontal shift of 2 units to the right is indicated by +2 inside the cubic root.
03

Apply the Transformations

The negative before the x causes each point (x,y) on the original function to be transformed to the point (-x,y), resulting in a reflection of the graph about the y-axis. The +2 under the cube root shifts each point to the right by 2 units, transforming each point from (x,y) to (x+2,y).
04

Graph the Transformations

Once the transformations are calculated, apply them to the graph of the original function. This will result in a graph of the new function \(g(x)=\sqrt[3]{-x+2}\).

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Most popular questions from this chapter

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?

The regular price of a computer is \(x\) dollars. Let \(f(x)=x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the computer, \(f \circ g\) or \(g \circ f ?\) Explain.

Solve each quadratic equation by the method of your choice. $$-x^{2}-2 x+1=0$$

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\sqrt{x}, g(x)=x-3$$

Solve and check: \(\frac{x-1}{5}-\frac{x+3}{2}=1-\frac{x}{4}\) (Section \(1.2, \text { Example } 3)\)
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