Question: The function g is defined by \(\mathrm{g(x) = 10 - \sqrt{x}}\). If \(\mathrm{g(c) = 4}\) for some value c,...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = 10 - \sqrt{x}}\)
  • Condition: \(\mathrm{g(c) = 4}\)
  • Need to find: the value of c

2. TRANSLATE the condition into an equation

• Since \(\mathrm{g(c) = 4}\), we substitute c into our function:
  \(\mathrm{g(c) = 10 - \sqrt{c} = 4}\)
• This gives us the equation: \(\mathrm{10 - \sqrt{c} = 4}\)

3. SIMPLIFY by isolating the square root term

• Subtract 10 from both sides:
  \(\mathrm{10 - \sqrt{c} - 10 = 4 - 10}\)
  \(\mathrm{-\sqrt{c} = -6}\)
• Multiply both sides by -1 to make the square root positive:
  \(\mathrm{\sqrt{c} = 6}\)

4. SIMPLIFY by squaring both sides

• Since \(\mathrm{\sqrt{c} = 6}\), square both sides:
  \(\mathrm{(\sqrt{c})^2 = 6^2}\)
  \(\mathrm{c = 36}\)

5. Verify the answer

• Check: \(\mathrm{g(36) = 10 - \sqrt{36} = 10 - 6 = 4}\) ✓

Answer: 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign confusion when handling "\(\mathrm{-\sqrt{c} = -6}\)"

Students often get confused by the negative signs and might:

• Leave the equation as "\(\mathrm{-\sqrt{c} = -6}\)" and try to square both sides directly, getting \(\mathrm{c = 36}\) but through incorrect reasoning
• Make an error like \(\mathrm{\sqrt{c} = -6}\), leading them to think there's no solution since square roots can't be negative
• Forget to multiply by -1, leaving them with \(\mathrm{\sqrt{c} = -6}\) and concluding no real solution exists

This leads to confusion and potential guessing among answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Not properly setting up the initial equation

Some students might misinterpret the function notation and set up incorrect equations like:

• Writing \(\mathrm{g(c) = 10 - \sqrt{c} = c}\) instead of \(\mathrm{g(c) = 4}\)
• Confusing the relationship between the function and the given condition

This causes them to work with the wrong equation entirely and arrive at an incorrect answer.

The Bottom Line:

This problem tests whether students can systematically work through function substitution while carefully managing algebraic manipulations involving square roots and negative signs. The key is staying organized and checking each step for sign errors.