If \(\mathrm{f(x) = x + 7}\) and \(\mathrm{g(x) = 7x}\), what is the value of \(\mathrm{4f(2) - g(2)}\)?

1. TRANSLATE the expression into specific calculations

• Given information:
  • \(\mathrm{f(x) = x + 7}\)
  • \(\mathrm{g(x) = 7x}\)
  • Need to find: \(\mathrm{4f(2) - g(2)}\)

• This tells us we need to:
  • Find \(\mathrm{f(2)}\) by substituting \(\mathrm{x = 2}\) into \(\mathrm{f(x)}\)
  • Find \(\mathrm{g(2)}\) by substituting \(\mathrm{x = 2}\) into \(\mathrm{g(x)}\)
  • Calculate 4 times \(\mathrm{f(2)}\), then subtract \(\mathrm{g(2)}\)

2. INFER the solution sequence

• We must evaluate the functions first before we can perform the final calculation
• The order matters: function evaluation → multiplication → subtraction

3. Evaluate f(2)

\(\mathrm{f(2) = 2 + 7 = 9}\)

4. Evaluate g(2)

\(\mathrm{g(2) = 7(2) = 14}\)

5. SIMPLIFY the final expression

\(\mathrm{4f(2) - g(2) = 4(9) - 14}\)
\(\mathrm{= 36 - 14}\)
\(\mathrm{= 22}\)

Answer: C. 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the expression "\(\mathrm{4f(2) - g(2)}\)" and try to work with the original functions instead of evaluating them first.

For example, they might write: \(\mathrm{4(x + 7) - 7x}\) and then try to substitute \(\mathrm{x = 2}\), getting \(\mathrm{4(2 + 7) - 7(2) = 4(9) - 14 = 22}\). While this actually leads to the correct answer, it shows they didn't understand the notation properly.

More problematically, some students get confused about what "\(\mathrm{4f(2)}\)" means and calculate \(\mathrm{f(4×2) = f(8)}\) instead of \(\mathrm{4×f(2)}\).

This leads to confusion and incorrect calculations.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly evaluate \(\mathrm{f(2) = 9}\) and \(\mathrm{g(2) = 14}\), but make arithmetic errors in the final step.

Common mistakes include:

• \(\mathrm{4(9) - 14 = 36 - 14 = 24}\) (addition instead of subtraction)
• \(\mathrm{4(9) - 14 = 4(9 - 14) = 4(-5) = -20}\) (incorrect order of operations)

This may lead them to select Choice A (-5) or get an answer not among the choices and guess.

The Bottom Line:

This problem tests whether students can correctly interpret function notation and maintain careful arithmetic through multiple steps. The key is recognizing that function evaluation must happen before the final arithmetic operations.