The function f is defined by \(\mathrm{f(x) = 270 × 3^{x-2} × (1/3)^{2-x}}\). What is the value of \(\mathrm{f(2)}\)?03090270810

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{f(x) = 270 × 3^{x-2} × (1/3)^{2-x}}\)
  • Need to find \(\mathrm{f(2)}\)

• What this tells us: We need to substitute \(\mathrm{x = 2}\) into the function expression

2. SIMPLIFY by substituting x = 2

• Replace every x with 2:

\(\mathrm{f(2) = 270 × 3^{2-2} × (1/3)^{2-2}}\)

• Calculate the exponents:
  • \(\mathrm{2 - 2 = 0}\)
  • \(\mathrm{2 - 2 = 0}\)

So: \(\mathrm{f(2) = 270 × 3^0 × (1/3)^0}\)

3. SIMPLIFY using the zero exponent rule

• Apply the rule \(\mathrm{a^0 = 1}\) for any nonzero number:
  • \(\mathrm{3^0 = 1}\)
  • \(\mathrm{(1/3)^0 = 1}\)

• Final calculation:

\(\mathrm{f(2) = 270 × 1 × 1 = 270}\)

Answer: D. 270

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about zero exponents: Many students incorrectly believe that \(\mathrm{3^0 = 0}\) instead of \(\mathrm{3^0 = 1}\).

With this misconception, they calculate:

\(\mathrm{f(2) = 270 × 0 × 0 = 0}\)

This leads them to select Choice A (0).

Second Most Common Error:

Weak SIMPLIFY execution: Students might think that \(\mathrm{3^0 = 3}\), forgetting that any nonzero number raised to the zero power equals 1.

This gives them \(\mathrm{f(2) = 270 × 3 × 3 = 2,430}\), which isn't among the answer choices. This causes them to get stuck and guess randomly.

The Bottom Line:

The zero exponent rule (\(\mathrm{a^0 = 1}\)) is often the most forgotten or misunderstood exponent rule. Students who don't have this rule firmly memorized will struggle with this otherwise straightforward substitution problem.