The function g is defined by \(\mathrm{g(x) = 2^x + 9}\). For which value of x is \(\mathrm{g(x) = 41}\)?...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 2^x + 9}\) and we need \(\mathrm{g(x) = 41}\)
• This means: \(\mathrm{2^x + 9 = 41}\)

2. SIMPLIFY to isolate the exponential term

• Subtract 9 from both sides: \(\mathrm{2^x + 9 - 9 = 41 - 9}\)
• This gives us: \(\mathrm{2^x = 32}\)

3. INFER the value of x

• We need to find what power of 2 equals 32
• Think through powers of 2: \(\mathrm{2^1 = 2}\), \(\mathrm{2^2 = 4}\), \(\mathrm{2^3 = 8}\), \(\mathrm{2^4 = 16}\), \(\mathrm{2^5 = 32}\)
• Therefore: \(\mathrm{x = 5}\)

Answer: A (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not readily recognize powers of 2 or make calculation errors when determining what power gives 32.

For example, they might incorrectly think \(\mathrm{2^4 = 32}\) or get confused with the sequence of powers, leading them to select Choice C (4) or become uncertain and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Some students may misinterpret the function notation and try to solve \(\mathrm{2^x = 41}\) instead of \(\mathrm{2^x + 9 = 41}\).

This leads to trying to find what power of 2 equals 41, which doesn't correspond to any simple power of 2, causing confusion and likely guessing among the answer choices.

The Bottom Line:

This problem tests both algebraic manipulation skills and knowledge of basic exponential relationships. Success requires systematically isolating the exponential term and recognizing common powers of 2.