Question:The function h is defined by \(\mathrm{h(x) = |x - 4| + 3}\). What is the value of \(\mathrm{h(1)}\)?0368

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = |x - 4| + 3}\)
  • Need to find: \(\mathrm{h(1)}\)
• This means substitute \(\mathrm{x = 1}\) into the function

2. TRANSLATE the substitution

• Replace every x in \(\mathrm{h(x) = |x - 4| + 3}\) with the value 1
• \(\mathrm{h(1) = |1 - 4| + 3}\)

3. SIMPLIFY inside the absolute value first

• Calculate \(\mathrm{1 - 4 = -3}\)
• \(\mathrm{h(1) = |-3| + 3}\)

4. SIMPLIFY by evaluating the absolute value

• Remember: absolute value gives the distance from zero
• \(\mathrm{|-3| = 3}\) (the distance from -3 to 0 is 3 units)
• \(\mathrm{h(1) = 3 + 3}\)

5. SIMPLIFY the final arithmetic

• \(\mathrm{h(1) = 3 + 3 = 6}\)

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with absolute value: Students correctly substitute and get \(\mathrm{|-3| + 3}\), but then evaluate \(\mathrm{|-3|}\) as -3 instead of 3.

They think "the absolute value of negative 3 is still negative 3" instead of remembering that absolute value always gives a positive result (or zero). This gives them \(\mathrm{h(1) = -3 + 3 = 0}\).

This leads them to select Choice A (0).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what \(\mathrm{h(1)}\) means and try to solve \(\mathrm{|x - 4| + 3 = 1}\) instead of substituting \(\mathrm{x = 1}\) into the function.

This creates confusion about what they're supposed to be finding, leading to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students truly understand both function notation (what it means to evaluate a function at a specific value) and the definition of absolute value (always non-negative). The combination of these two concepts in one problem creates multiple opportunities for conceptual confusion.