The function is defined by . For what value of does ?

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SAT Nonlinear Functions Question 193 | Free SAT Prep | Aniko

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Question 193·Easy·Nonlinear Functions

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The function $g$ is defined by $g(x)=2\sqrt[3]{x}-1$ . For what value of $x$ does $g(x)=5$ ?

For cube-root and other radical equations on the SAT, first substitute into the given function definition to write a single equation, then isolate the radical term step by step (undo addition/subtraction, then multiplication/division). Once the radical is alone, apply the inverse operation (square both sides for a square root, cube both sides for a cube root, etc.) and then evaluate the resulting simple power carefully, watching out not to confuse squaring with cubing or to carry along coefficients that you already removed.

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Write the equation

Substitute $g(x)=2\sqrt[3]{x}-1$ into the condition $g(x)=5$ . What equation do you get involving $x$ ?

Isolate the cube root

Your equation should have $2\sqrt[3]{x}-1$ on one side and a number on the other. What two steps will isolate $\sqrt[3]{x}$ on one side of the equation?

Undo the cube root

Once you have an equation of the form $\sqrt[3]{x} = \text{(number)}$ , what operation will undo the cube root and give you $x$ ?

Compute carefully

After you undo the cube root, you will get a power of 3. Make sure you are raising 3 to the correct power, not squaring it.

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Graph both sides of the equation

In one expression line, type y = 2*x^(1/3) - 1. In another line, type y = 5. Zoom out if needed and look for the point where the two graphs intersect; the x-coordinate of that intersection is the solution to $g(x)=5$ .

Step-by-step Explanation

Set up the equation

The function is $g(x)=2\sqrt[3]{x}-1$ . We are told $g(x)=5$ , so write the equation:

2\sqrt[3]{x} - 1 = 5

Isolate the cube root term

Undo the $-1$ by adding 1 to both sides:

2\sqrt[3]{x} = 6

Now divide both sides by 2 to isolate the cube root:

\sqrt[3]{x} = 3

Undo the cube root

To remove a cube root, cube both sides of the equation. Cubing is the inverse (opposite) of taking a cube root:

(\sqrt[3]{x})^3 = 3^3

This simplifies to:

x = 3^3

Evaluate and choose the answer

Compute $3^3$ :

3^3 = 27

So $x=27$ , which corresponds to choice C.

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Step-by-step Explanation

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Step-by-step Explanation