\(\mathrm{f(x) = x + b}\). For the linear function f, b is a constant. When x = 0, \(\mathrm{f(x) =...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = x + b}\) (linear function with constant b)
  • When \(\mathrm{x = 0}\), \(\mathrm{f(x) = 30}\)

• What this tells us: We need to use the specific point \(\mathrm{(0, 30)}\) that lies on this linear function to find b

2. TRANSLATE the condition into mathematical form

• The condition "when x = 0, f(x) = 30" means:
  • We substitute \(\mathrm{x = 0}\) into our function
  • The output should equal 30
  • So: \(\mathrm{f(0) = 30}\)

3. SIMPLIFY by substitution and solving

• Substitute \(\mathrm{x = 0}\) into \(\mathrm{f(x) = x + b}\):
  \(\mathrm{f(0) = 0 + b = b}\)

• Since \(\mathrm{f(0) = 30}\):
  \(\mathrm{b = 30}\)

Answer: D. 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle with function notation and not understand that "when x = 0, f(x) = 30" means they need to substitute these values into the function equation.

Instead of recognizing that \(\mathrm{f(0) = 30}\) means \(\mathrm{0 + b = 30}\), they might think they need to solve a more complex equation or might confuse the direction of the relationship. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can connect the abstract function notation with concrete numerical relationships. The key insight is recognizing that a specific input-output pair gives us exactly the information needed to find the unknown parameter.