For the linear function h, \(\mathrm{h(x) = x + b}\), b is a constant and \(\mathrm{h(0) = 45}\). What is...

1. TRANSLATE the problem information

• Given information:
  • Linear function: \(\mathrm{h(x) = x + b}\)
  • b is a constant (unknown we need to find)
  • \(\mathrm{h(0) = 45}\) (when input is 0, output is 45)

• What we need to find: The value of b

2. INFER the solution approach

• Key insight: \(\mathrm{h(0) = 45}\) tells us what happens when we substitute \(\mathrm{x = 0}\) into our function
• Strategy: Substitute \(\mathrm{x = 0}\) into \(\mathrm{h(x) = x + b}\) and use the fact that the result equals 45

3. Substitute and solve

• When \(\mathrm{x = 0}\) in \(\mathrm{h(x) = x + b}\):
  \(\mathrm{h(0) = 0 + b = b}\)

• Since we know \(\mathrm{h(0) = 45}\):
  \(\mathrm{b = 45}\)

Answer: 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what \(\mathrm{h(0) = 45}\) means in terms of the function

Students might see \(\mathrm{h(0) = 45}\) as just another piece of information rather than understanding it means "when I put 0 into the function, I get 45 out." They may try to set up more complex equations or get confused about what they're supposed to do with this information.

This leads to confusion and guessing rather than systematic solution.

The Bottom Line:

This problem tests whether students truly understand function notation. The mathematical work is minimal once you realize that \(\mathrm{h(0)}\) simply means "substitute 0 for x in the function." The challenge is making that conceptual connection between the notation and the substitution process.