\(\mathrm{f(x) = b^x}\). For the exponential function f, b is a positive constant. When x = 2, \(\mathrm{f(x) = 9}\)....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = b^x}\) (exponential function)
  • b is a positive constant
  • When \(\mathrm{x = 2}\), \(\mathrm{f(x) = 9}\)
• This tells us we need to find what value of b makes \(\mathrm{f(2) = 9}\)

2. TRANSLATE the key condition into an equation

• 'When \(\mathrm{x = 2}\), \(\mathrm{f(x) = 9}\)' means \(\mathrm{f(2) = 9}\)
• Since \(\mathrm{f(x) = b^x}\), we have \(\mathrm{f(2) = b^2}\)
• Therefore: \(\mathrm{b^2 = 9}\)

3. SIMPLIFY to solve for b

• Take the square root of both sides: \(\mathrm{b = ±\sqrt{9}}\)
• Evaluate: \(\mathrm{b = ±3}\)
• Since the problem states b is positive: \(\mathrm{b = 3}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to connect 'when \(\mathrm{x = 2}\), \(\mathrm{f(x) = 9}\)' with the need to substitute \(\mathrm{x = 2}\) into the function formula. Instead of setting up \(\mathrm{f(2) = b^2 = 9}\), they might try to work backwards from the answer choices or set up incorrect equations.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{b^2 = 9}\) but make arithmetic errors when taking the square root. They might incorrectly think \(\mathrm{\sqrt{9} = 4.5}\) (perhaps confusing it with \(\mathrm{9/2}\)) or make other computational mistakes.

This may lead them to select Choice B (4.5).

The Bottom Line:

This problem tests whether students can translate a function condition into a solvable equation and then execute basic algebraic operations accurately. The key insight is recognizing that '\(\mathrm{f(2) = 9}\)' directly gives you an equation in terms of b.