\(\mathrm{h(x) = 2x + 3}\)For the function h defined above, what is the value of \(\mathrm{h(3a) - h(a)}\) when a...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = 2x + 3}\)
  • Need to find: \(\mathrm{h(3a) - h(a)}\) when \(\mathrm{a = 2}\)

• What this tells us: We need to evaluate the function at two different input values, then subtract

2. TRANSLATE the specific values

• Since \(\mathrm{a = 2}\):
  • \(\mathrm{h(3a)}\) becomes \(\mathrm{h(3 \times 2) = h(6)}\)
  • \(\mathrm{h(a)}\) becomes \(\mathrm{h(2)}\)

• Now we have: \(\mathrm{h(6) - h(2)}\)

3. SIMPLIFY by evaluating each function

• Evaluate \(\mathrm{h(6)}\):
  \(\mathrm{h(6) = 2(6) + 3}\)
  \(\mathrm{= 12 + 3}\)
  \(\mathrm{= 15}\)

• Evaluate \(\mathrm{h(2)}\):
  \(\mathrm{h(2) = 2(2) + 3}\)
  \(\mathrm{= 4 + 3}\)
  \(\mathrm{= 7}\)

4. SIMPLIFY the final calculation

• \(\mathrm{h(3a) - h(a)}\)
  \(\mathrm{= h(6) - h(2)}\)
  \(\mathrm{= 15 - 7}\)
  \(\mathrm{= 8}\)

Answer: B (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try to substitute \(\mathrm{a = 2}\) directly into the function before identifying what \(\mathrm{h(3a)}\) and \(\mathrm{h(a)}\) represent. They might write something like \(\mathrm{h(3a) = 2(3a) + 3}\)
\(\mathrm{= 6a + 3}\)
\(\mathrm{= 6(2) + 3}\)
\(\mathrm{= 15}\), then get confused about what to do next.

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need \(\mathrm{h(6)}\) and \(\mathrm{h(2)}\), but make arithmetic errors in the function evaluations or final subtraction. For example, calculating \(\mathrm{h(6) = 2(6) + 3 = 14}\) instead of 15, leading to a final answer of 7.

This may lead them to select an incorrect answer choice or realize their error and guess.

The Bottom Line:

This problem tests whether students can handle the translation from algebraic expressions with variables to specific numerical evaluations. The key insight is recognizing that you must first substitute the given value of a to determine the specific inputs to the function, then evaluate each function call separately.