The function h is defined by \(\mathrm{h(x) = \frac{8}{5x+6}}\). What is the value of \(\mathrm{h(2)}\)?

1. TRANSLATE the question

• Given information:
  • Function definition: \(\mathrm{h(x) = \frac{8}{5x+6}}\)
  • Need to find: \(\mathrm{h(2)}\)

• What this tells us: We need to substitute \(\mathrm{x = 2}\) into the function formula

2. SIMPLIFY through substitution and calculation

• Replace x with 2 in the formula:
  \(\mathrm{h(2) = \frac{8}{5(2)+6}}\)

• Follow order of operations in the denominator first:
  • Calculate \(\mathrm{5(2) = 10}\)
  • Add: \(\mathrm{10 + 6 = 16}\)
  • So \(\mathrm{h(2) = \frac{8}{16}}\)

• Simplify the fraction:
  \(\mathrm{\frac{8}{16} = \frac{1}{2}}\)

Answer: \(\mathrm{\frac{1}{2}}\) (which equals \(\mathrm{0.5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make order of operations errors, particularly calculating \(\mathrm{5x + 6}\) incorrectly when \(\mathrm{x = 2}\).

For example, they might calculate \(\mathrm{5(2) + 6}\) as \(\mathrm{5(8) = 40}\), thinking they multiply 5 by \(\mathrm{(2+6)}\) first. This gives \(\mathrm{h(2) = \frac{8}{40} = \frac{1}{5}}\) instead of the correct \(\mathrm{\frac{1}{2}}\). This leads to confusion since \(\mathrm{\frac{1}{5}}\) or \(\mathrm{0.2}\) doesn't match typical answer formats.

The Bottom Line:

This problem tests whether students can systematically substitute values into functions and maintain proper order of operations - a fundamental skill that becomes critical in more complex function work.