If f is the function defined by \(\mathrm{f(x) = \frac{2x - 1}{3}}\), what is the value of \(\mathrm{f(5)}\)?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = \frac{2x - 1}{3}}\)
  • Need to find \(\mathrm{f(5)}\)
• What this tells us: We need to substitute \(\mathrm{x = 5}\) into the function

2. TRANSLATE the substitution

• Replace every x in the function with 5:
  • \(\mathrm{f(5) = \frac{2(5) - 1}{3}}\)

3. SIMPLIFY the expression step by step

• First, multiply: \(\mathrm{2(5) = 10}\)
  • \(\mathrm{f(5) = \frac{10 - 1}{3}}\)
• Next, subtract in the numerator: \(\mathrm{10 - 1 = 9}\)
  • \(\mathrm{f(5) = \frac{9}{3}}\)
• Finally, divide: \(\mathrm{9/3 = 3}\)
  • \(\mathrm{f(5) = 3}\)

Answer: C. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step calculation process.

For example, they might:

• Forget to multiply \(\mathrm{2 \times 5}\) first, getting \(\mathrm{\frac{2 + 5 - 1}{3} = \frac{6}{3} = 2}\)
• Only evaluate the numerator and stop: \(\mathrm{2(5) - 1 = 9}\), selecting Choice D (9)
• Make order of operations errors like calculating \(\mathrm{2 \times 5/3}\) first, then subtracting 1

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students partially substitute or misinterpret the function notation.

They might substitute incorrectly, like treating \(\mathrm{f(5)}\) as \(\mathrm{f \times 5}\), or only substituting one instance of x, leading to confusion and potentially selecting Choice A (4/3) or Choice B (7/3).

The Bottom Line:

Function evaluation requires careful attention to substitution and systematic arithmetic - students who rush through the calculation steps often make preventable errors.