\(\mathrm{f(x) = 2(3^x)}\)For the function f defined above, what is the value of \(\mathrm{f(2)}\)?

1. TRANSLATE the function notation

• Given: \(\mathrm{f(x) = 2(3^x)}\)
• Need to find: \(\mathrm{f(2)}\)
• What this means: Replace every x in the expression with 2

2. SIMPLIFY through substitution

• Substitute \(\mathrm{x = 2}\): \(\mathrm{f(2) = 2(3^2)}\)
• Now we have a numerical expression to evaluate

3. SIMPLIFY using order of operations

• Evaluate the exponent first: \(\mathrm{3^2 = 9}\)
• Then multiply: \(\mathrm{f(2) = 2(9) = 18}\)

Answer: 18 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly treat the exponent as multiplication instead of repeated multiplication.

They might calculate \(\mathrm{2(3 \times 2) = 2(6) = 12}\), thinking that \(\mathrm{3^2}\) means "3 times 2" rather than "3 squared." This leads them to select Choice B (12).

Second Most Common Error:

Poor order of operations in SIMPLIFY: Students incorrectly distribute operations or forget the coefficient entirely.

Some students evaluate only \(\mathrm{3^2 = 9}\) and forget about the coefficient 2, leading them to select Choice A (9). Others might incorrectly calculate \(\mathrm{(2 \times 3)^2 = 6^2 = 36}\), leading to Choice D (36).

The Bottom Line:

This problem tests careful execution of function substitution and order of operations. The key is remembering that exponents represent repeated multiplication (\(\mathrm{3^2 = 3 \times 3}\)), and that you must include all parts of the original expression when substituting.