The function g is defined by \(\mathrm{g(x) = 150(2)^{(x-4)}}\). What is the value of \(\mathrm{g(4)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 150(2)^{(x-4)}}\)
  • Need to find: \(\mathrm{g(4)}\)
• This tells us we need to substitute \(\mathrm{x = 4}\) into the function

2. TRANSLATE the substitution process

• Replace every x in the function with 4:
  \(\mathrm{g(4) = 150(2)^{(4-4)}}\)

3. SIMPLIFY the exponent

• Work inside the parentheses first (order of operations):
  \(\mathrm{4 - 4 = 0}\)
• Now we have: \(\mathrm{g(4) = 150(2)^{0}}\)

4. INFER and apply the zero exponent rule

• Any non-zero number raised to the power of 0 equals 1
• Therefore: \(\mathrm{2^{0} = 1}\)
• Substitute back: \(\mathrm{g(4) = 150 \times 1}\)

5. SIMPLIFY the final calculation

• \(\mathrm{150 \times 1 = 150}\)

Answer: C) 150

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Zero exponent rule

Students often think \(\mathrm{2^{0} = 0}\) (confusing it with multiplication by zero) or \(\mathrm{2^{0} = 2}\) (thinking the exponent doesn't change the base). This leads them to calculate either \(\mathrm{g(4) = 150 \times 0 = 0}\) or \(\mathrm{g(4) = 150 \times 2 = 300}\).

This may lead them to select Choice A (0) or Choice D (300)

Second Most Common Error:

Weak TRANSLATE skill: Improper substitution

Some students substitute incorrectly, perhaps writing \(\mathrm{g(4) = 150(2)^{(x-4)}}\) without replacing x with 4, or they make arithmetic errors when calculating 4-4. This leads to confusion about what the exponent should be.

This causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests whether students remember the fundamental zero exponent rule and can properly substitute values into exponential functions. The key insight is recognizing that when x = 4, the exponent becomes zero, making the exponential part equal to 1.