\(\mathrm{g(x) = 5(2^{x-1})}\) For the function g defined above, what is the value of \(\mathrm{g(4)}\)? 8 20 40 80...

1. TRANSLATE the problem information

• Given: Function \(\mathrm{g(x) = 5(2^{(x-1)})}\)
• Find: The value of \(\mathrm{g(4)}\)
• This means substitute \(\mathrm{x = 4}\) into the function

2. TRANSLATE the substitution

• Replace every x with 4: \(\mathrm{g(4) = 5(2^{(4-1)})}\)

3. SIMPLIFY using order of operations

• First, handle the expression inside the exponent parentheses: \(\mathrm{4-1 = 3}\)
• Now we have: \(\mathrm{g(4) = 5(2^3)}\)
• Next, calculate the exponent: \(\mathrm{2^3 = 2 \times 2 \times 2 = 8}\)
• Now we have: \(\mathrm{g(4) = 5(8)}\)
• Finally, multiply: \(\mathrm{5 \times 8 = 40}\)

Answer: 40 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution - Order of operations confusion: Students might incorrectly multiply \(\mathrm{5 \times 2 = 10}\) first, then try to raise this to some power, rather than following the proper order of operations where exponents come before multiplication.

For example, they might calculate \(\mathrm{5 \times 2 = 10}\), then get confused about what to do with the exponent, leading to incorrect intermediate calculations that don't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor arithmetic in SIMPLIFY: Students correctly follow the order of operations but make calculation errors, such as computing \(\mathrm{2^3 = 6}\) instead of 8, which gives \(\mathrm{5 \times 6 = 30}\) (not among the choices), or making errors in the final multiplication step.

This causes them to get stuck since their calculated answer doesn't match any choice, leading to second-guessing their approach and random answer selection.

The Bottom Line:

This problem tests whether students can systematically apply function notation and order of operations. Success requires careful step-by-step execution rather than rushing through the arithmetic.