\(\mathrm{f(x) = 4x + b}\). For the linear function f, b is a constant and \(\mathrm{f(7) = 28}\). What is...

1. TRANSLATE the given information into mathematical equation

• Given information:
  • \(\mathrm{f(x) = 4x + b}\) (linear function with unknown constant b)
  • \(\mathrm{f(7) = 28}\) (when input is 7, output is 28)

• This tells us we can substitute \(\mathrm{x = 7}\) and set the result equal to 28

2. TRANSLATE the condition into a solvable equation

• Since \(\mathrm{f(7) = 28}\), substitute \(\mathrm{x = 7}\) into \(\mathrm{f(x) = 4x + b}\):
  \(\mathrm{f(7) = 4(7) + b = 28}\)

• This gives us: \(\mathrm{28 + b = 28}\)

3. SIMPLIFY to solve for b

• From \(\mathrm{28 + b = 28}\)
• Subtract 28 from both sides: \(\mathrm{b = 28 - 28 = 0}\)

Answer: A. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which value represents what in the function notation. They might think b should equal 7 (the input value) or 4 (the coefficient), rather than understanding that \(\mathrm{f(7) = 28}\) means 'when \(\mathrm{x = 7}\), the entire expression \(\mathrm{4x + b}\) equals 28.'

This conceptual confusion may lead them to select Choice D (7) or Choice C (4) without performing the substitution correctly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{28 = 4(7) + b}\) but make algebraic errors. They might incorrectly think that since \(\mathrm{4(7) = 28}\), then b must equal 1 to make the equation 'work,' misunderstanding that \(\mathrm{28 + b = 28}\) means \(\mathrm{b = 0}\).

This may lead them to select Choice B (1).

The Bottom Line:

This problem tests whether students truly understand function notation and can translate a functional relationship into an algebraic equation. The key insight is recognizing that \(\mathrm{f(7) = 28}\) creates a direct substitution opportunity, not a separate relationship to memorize.