For the function f defined by \(\mathrm{f(x) = \frac{k}{x}}\), where k is a constant, if \(\mathrm{f(4) = 15}\), what is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = \frac{k}{x}}\) (rational function with unknown constant k)
  • \(\mathrm{f(4) = 15}\) (function value when x = 4)
• What this tells us: We need to find the value of constant k

2. TRANSLATE the condition into an equation

• Since \(\mathrm{f(4) = 15}\), substitute \(\mathrm{x = 4}\) into \(\mathrm{f(x) = \frac{k}{x}}\):

\(\mathrm{f(4) = \frac{k}{4} = 15}\)

• This gives us the equation: \(\mathrm{\frac{k}{4} = 15}\)

3. INFER the solution strategy

• To find k, we need to isolate it on one side of the equation
• Since k is divided by 4, multiply both sides by 4

4. SIMPLIFY to find k

• \(\mathrm{\frac{k}{4} = 15}\)
• Multiply both sides by 4: \(\mathrm{k = 15 \times 4 = 60}\)

Answer: D) 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusion about function notation and substitution

Students might think \(\mathrm{f(4) = 15}\) means "4 divided by k equals 15" instead of "k divided by 4 equals 15." This leads them to set up the equation as \(\mathrm{\frac{4}{k} = 15}\), giving \(\mathrm{k = \frac{4}{15}}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes

Students correctly set up \(\mathrm{\frac{k}{4} = 15}\) but make calculation errors. For example, they might compute \(\mathrm{15 \times 4 = 56}\) or \(\mathrm{15 \times 4 = 40}\), leading them to select incorrect answer choices or abandon their systematic approach.

The Bottom Line:

This problem tests whether students truly understand function notation and can correctly substitute values. The mathematics is straightforward once the setup is correct, but the function notation can be a stumbling block for students who haven't solidified this fundamental concept.