If \(\mathrm{h(x) = x^2 - 3}\) and \(\mathrm{k(x) = 2^x}\), what is the value of \(\mathrm{3h(4) + k(3)}\)?

1. TRANSLATE the problem requirements

• Given information:
  • \(\mathrm{h(x) = x^2 - 3}\)
  • \(\mathrm{k(x) = 2^x}\)
  • Need to find: \(\mathrm{3h(4) + k(3)}\)

• This tells us we need to:
  • Substitute 4 into function h
  • Substitute 3 into function k
  • Multiply the result of \(\mathrm{h(4)}\) by 3
  • Add both results together

2. SIMPLIFY by evaluating h(4)

• Substitute \(\mathrm{x = 4}\) into \(\mathrm{h(x) = x^2 - 3}\):
  \(\mathrm{h(4) = 4^2 - 3 = 16 - 3 = 13}\)

3. SIMPLIFY by evaluating k(3)

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{k(x) = 2^x}\):
  \(\mathrm{k(3) = 2^3 = 8}\)

4. SIMPLIFY the final expression

• Calculate \(\mathrm{3h(4)}\): \(\mathrm{3(13) = 39}\)
• Calculate \(\mathrm{3h(4) + k(3)}\): \(\mathrm{39 + 8 = 47}\)

Answer: C (47)

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students correctly evaluate both functions but forget to complete the final addition step.

They calculate \(\mathrm{3h(4) = 39}\) correctly but stop there, selecting this as their final answer without adding \(\mathrm{k(3) = 8}\).

This may lead them to select Choice B (39).

Second Most Common Error:

Conceptual confusion about exponential notation: Students misinterpret \(\mathrm{k(3) = 2^3}\) as multiplication instead of exponentiation.

They calculate \(\mathrm{k(3) = 2 \times 3 = 6}\) instead of \(\mathrm{k(3) = 2^3 = 8}\), then compute \(\mathrm{3h(4) + k(3) = 39 + 6 = 45}\). Since 45 isn't among the choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically work through multi-step function evaluation while maintaining attention to detail. The key is completing every operation in the expression, not just the ones that seem most obvious.