The function f is defined by \(\mathrm{f(x) = 8\sqrt{x}}\). For what value of x does \(\mathrm{f(x) = 48}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 8\sqrt{x}}\)
  • Condition: \(\mathrm{f(x) = 48}\)
• This tells us we need to solve: \(\mathrm{8\sqrt{x} = 48}\)

2. SIMPLIFY to isolate the square root

• Divide both sides by 8:

\(\mathrm{8\sqrt{x} \div 8 = 48 \div 8}\)
\(\mathrm{\sqrt{x} = 6}\)

3. SIMPLIFY to solve for x

• Since \(\mathrm{\sqrt{x} = 6}\), we need to eliminate the square root
• Square both sides:

\(\mathrm{(\sqrt{x})^2 = 6^2}\)
\(\mathrm{x = 36}\)

4. Verify the answer

• Check: \(\mathrm{f(36) = 8\sqrt{36} = 8(6) = 48}\) ✓

Answer: C. 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly get to \(\mathrm{\sqrt{x} = 6}\) but then think this means \(\mathrm{x = 6}\), forgetting that they need to square both sides to eliminate the square root.

They see \(\mathrm{\sqrt{x} = 6}\) and mistakenly conclude that \(\mathrm{x = 6}\), not recognizing that the square root symbol means "what number, when squared, gives x." To find x itself, they must square the 6.

This leads them to select Choice A (6).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the problem setup and think they need to find when \(\mathrm{8x = 48}\), ignoring the square root entirely.

They solve \(\mathrm{8x = 48}\) to get \(\mathrm{x = 6}\), or they get confused about what operation to perform with the numbers 8 and 48, leading to \(\mathrm{x = 8}\).

This may lead them to select Choice A (6) or Choice B (8).

The Bottom Line:

This problem tests whether students understand that solving equations with square roots requires "undoing" the square root operation by squaring both sides. The key insight is recognizing that \(\mathrm{\sqrt{x} = 6}\) means x must equal \(\mathrm{6^2}\), not just 6.