The function f is defined by \(\mathrm{f(x) = 3\sqrt[3]{x}}\). For what value of x does \(\mathrm{f(x) = 18}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = 3\sqrt[3]{x}}\) and we need \(\mathrm{f(x) = 18}\)
• This means we need to solve: \(\mathrm{3\sqrt[3]{x} = 18}\)

2. INFER the solution strategy

• To solve for x, we need to isolate it step by step
• First isolate the cube root \(\mathrm{\sqrt[3]{x}}\), then eliminate the cube root operation

3. SIMPLIFY by isolating the cube root

• Divide both sides by 3: \(\mathrm{\sqrt[3]{x} = 6}\)

4. SIMPLIFY by eliminating the cube root

• Cube both sides: \(\mathrm{(\sqrt[3]{x})^3 = 6^3}\)
• This gives us: \(\mathrm{x = 216}\)

Answer: D) 216

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to cube both sides before isolating the cube root first.

They might write: \(\mathrm{(3\sqrt[3]{x})^3 = 18^3}\), leading to \(\mathrm{27x = 5,832}\), so \(\mathrm{x = 216}\). While this technically works, it's much more complex and error-prone. More commonly, they make mistakes in this approach and don't arrive at the right answer, leading to confusion and guessing.

Second Most Common Error:

Conceptual confusion about cube roots: Students confuse cube root with square root operations.

They might think \(\mathrm{\sqrt[3]{x} = 6}\) means \(\mathrm{x = 6}\) (treating it like a simple equation rather than recognizing the cube root). This leads them to select Choice A (6).

The Bottom Line:

This problem tests whether students can systematically work backwards from a function equation, using proper order of operations to isolate the variable step by step.