The function h is defined by \(\mathrm{h(x) = x^2 + 2x - 3}\). What is the value of \(\mathrm{h(5)}\)?

1. TRANSLATE the question requirements

• Given information:
  • Function: \(\mathrm{h(x) = x^2 + 2x - 3}\)
  • Need to find: \(\mathrm{h(5)}\)
• What this means: Substitute \(\mathrm{x = 5}\) into the function and calculate the result

2. SIMPLIFY through substitution and calculation

• Replace every x in the function with 5:
  \(\mathrm{h(5) = (5)^2 + 2(5) - 3}\)

• Follow order of operations (PEMDAS):
  • First, calculate the exponent: \(\mathrm{(5)^2 = 25}\)
  • Next, multiply: \(\mathrm{2(5) = 10}\)
  • Then combine: \(\mathrm{h(5) = 25 + 10 - 3}\)
  • Finally, add and subtract from left to right: \(\mathrm{25 + 10 = 35}\), then \(\mathrm{35 - 3 = 32}\)

Answer: C) 32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make order of operations mistakes, particularly with the exponent.

Instead of calculating \(\mathrm{(5)^2 = 25}\) first, they might work left to right and calculate \(\mathrm{5^2 + 2(5)}\) as \(\mathrm{(5 + 2)(5) = 35}\), then subtract 3 to get 32 by coincidence, or they might calculate incorrectly as \(\mathrm{5 + 2(5) = 15}\), leading to \(\mathrm{15 - 3 = 12}\), which isn't among the choices, causing confusion and guessing.

Second Most Common Error:

Arithmetic calculation mistakes: Students correctly understand the substitution process but make simple arithmetic errors.

For example, they might calculate \(\mathrm{25 + 10 - 3 = 28}\) instead of 32, or miscalculate \(\mathrm{(5)^2}\) as 10 instead of 25, leading to \(\mathrm{10 + 10 - 3 = 17}\). Since these values don't match the answer choices exactly, this leads to confusion and random selection.

The Bottom Line:

This problem tests whether students can systematically substitute values into functions and perform multi-step calculations accurately. Success requires careful attention to order of operations and arithmetic precision.