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Chapter 2: Problem 107

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
To graph \(g(x)=\sqrt[3]{x}+2\) we take original function \(f(x) = \sqrt[3]{x}\) and vertically shift it upward by 2 units.

Step by step solution

01

Sketch the Original Function

Start sketching the original cube root function, \(f(x) = \sqrt[3]{x}\). Make sure to note key feature: passing through origin and behaviour on positive and negative axis.
02

Understanding the Transformation

The function \(g(x)=\sqrt[3]{x}+2\) is a vertical shift of \(f(x) = \sqrt[3]{x}\). The '+2' adds a shift of 2 units up.
03

Graphing the Transformed Function

Use the transformations discussed to sketch \(g(x) = \sqrt[3]{x} + 2\) by shifting the graph of \(f(x) = \sqrt[3]{x}\) two units upward. The characteristics of the function will be the same as \(f(x) = \sqrt[3]{x}\), but the entire graph will be 2 units above the original graph.

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Most popular questions from this chapter

In Exercises \(105-108,\) you will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x .\) Then find and interpret \(S(30)\)

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+12 x-6 y-4=0 $$

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+2)^{2}+(y+2)^{2}=4 $$

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}-10 x-6 y-30=0 $$

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?
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