The function f is defined by \(\mathrm{f(x) = \frac{16}{x}}\). What is the value of \(\mathrm{f(x)}\) when x = 17?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{f(x) = \frac{16}{x}}\)
  • Need to find: \(\mathrm{f(x)}\) when \(\mathrm{x = 17}\)

• This means we need to find \(\mathrm{f(17)}\)

2. INFER the solution approach

• To evaluate any function at a specific value, substitute that value for the variable
• Since we want \(\mathrm{f(17)}\), we substitute 17 for x in the function \(\mathrm{f(x) = \frac{16}{x}}\)

3. Perform the substitution

• \(\mathrm{f(17) = \frac{16}{17}}\)

Answer: A. 16/17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may reverse the fraction, thinking that "\(\mathrm{x = 17}\)" means the 17 should go in the numerator position, leading them to write \(\mathrm{f(17) = \frac{17}{16}}\) instead of \(\mathrm{f(17) = \frac{16}{17}}\).

This may lead them to select Choice B (17/16)

Second Most Common Error:

Missing conceptual knowledge of function evaluation: Students might not understand that \(\mathrm{f(17)}\) requires substitution and instead think the answer is simply one of the given numbers (16 or 17) from the original function or problem.

This may lead them to select Choice C (16) or Choice D (17)

The Bottom Line:

This problem tests the fundamental skill of function evaluation through substitution. Success depends on correctly translating the question into mathematical notation and understanding that function evaluation means "replace the variable with the given value."