The function h is defined by \(\mathrm{h(x) = \frac{x}{3} + 2}\). For what value of x does \(\mathrm{h(x) = 7}\)?5152127

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = \frac{x}{3} + 2}\)
  • We need to find x when \(\mathrm{h(x) = 7}\)
• What this tells us: We need to set the function expression equal to 7 and solve for x

2. TRANSLATE the question into an equation

• "For what value of x does h(x) = 7?" becomes:
  \(\mathrm{\frac{x}{3} + 2 = 7}\)
• This equation represents the core relationship we need to solve

3. SIMPLIFY through algebraic manipulation

• Subtract 2 from both sides to isolate the term with x:
  \(\mathrm{\frac{x}{3} = 5}\)
• Multiply both sides by 3 to solve for x:
  \(\mathrm{x = 15}\)

4. Verify the solution

• Substitute x = 15 back into the original function:
  \(\mathrm{h(15) = \frac{15}{3} + 2 = 5 + 2 = 7}\) ✓

Answer: B) 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret what "h(x) = 7" means, thinking they need to substitute 7 for x in the original function instead of setting the function equal to 7.

They might calculate \(\mathrm{h(7) = \frac{7}{3} + 2 = 2.33... + 2 = 4.33...}\), then get confused when this doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{x}{3} + 2 = 7}\) but make arithmetic errors in the solving process. Common mistakes include:

• Subtracting 2 incorrectly: getting \(\mathrm{\frac{x}{3} = 9}\) instead of \(\mathrm{\frac{x}{3} = 5}\)
• Multiplying by 3 incorrectly: getting \(\mathrm{x = 27}\) instead of \(\mathrm{x = 15}\)

This may lead them to select Choice D) (27) if they made the subtraction error.

The Bottom Line:

This problem tests whether students understand the relationship between function notation and equation solving. The key insight is recognizing that "h(x) = 7" creates an equation where the function expression equals 7, not where x equals 7.