The function f is defined by \(\mathrm{f(x) = 4x^{-1}}\). What is the value of \(\mathrm{f(21)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function f is defined by \(\mathrm{f(x) = 4x^{-1}}\)
  • Need to find \(\mathrm{f(21)}\)
• This means: substitute 21 for x in the function

2. INFER the approach

• To find \(\mathrm{f(21)}\), I need to replace every x in the function with 21
• This gives me: \(\mathrm{f(21) = 4(21)^{-1}}\)
• The negative exponent tells me I need to apply the negative exponent rule

3. SIMPLIFY using the negative exponent rule

• Apply the rule: \(\mathrm{x^{-n} = \frac{1}{x^n}}\)
• So \(\mathrm{(21)^{-1} = \frac{1}{21}}\)
• Therefore: \(\mathrm{f(21) = 4 \times \frac{1}{21} = \frac{4}{21}}\)

Answer: \(\mathrm{\frac{4}{21}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative exponents: Students might think that a negative exponent makes the entire result negative, so they calculate \(\mathrm{21^{-1} = -21}\) instead of \(\mathrm{\frac{1}{21}}\). This gives them \(\mathrm{f(21) = 4 \times (-21) = -84}\).

This may lead them to select Choice A (\(\mathrm{-84}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret the function structure and think they need to find \(\mathrm{(4 \times 21)^{-1}}\) instead of \(\mathrm{4 \times (21)^{-1}}\). This gives them \(\mathrm{f(21) = 84^{-1} = \frac{1}{84}}\).

This may lead them to select Choice B (\(\mathrm{\frac{1}{84}}\)).

The Bottom Line:

The key challenge is correctly handling negative exponents - remembering that \(\mathrm{x^{-1}}\) means "one over x," not "negative x." Function evaluation itself is straightforward once you master this exponent rule.