If the function h is defined by \(\mathrm{h(x) = \frac{2x + 2}{x - 1}}\), what is the value of \(\mathrm{h(3)}\)?

1. TRANSLATE the problem information

• Given: Function \(\mathrm{h(x) = \frac{2x + 2}{x - 1}}\)
• Need to find: \(\mathrm{h(3)}\)
• This means: Replace every x in the function with 3

2. TRANSLATE the substitution

• Set up the substitution:

\(\mathrm{h(3) = \frac{2(3) + 2}{3 - 1}}\)

3. SIMPLIFY the numerator

• Follow order of operations - multiplication first:

\(\mathrm{2(3) + 2 = 6 + 2 = 8}\)

4. SIMPLIFY the denominator

• Evaluate: \(\mathrm{3 - 1 = 2}\)

5. SIMPLIFY the final division

• \(\mathrm{h(3) = \frac{8}{2} = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make order of operations errors when evaluating the numerator.

Instead of computing \(\mathrm{2(3) + 2 = 6 + 2 = 8}\), they might incorrectly calculate \(\mathrm{2(3 + 2) = 2(5) = 10}\), treating the addition as if it were inside parentheses. This gives them \(\mathrm{h(3) = \frac{10}{2} = 5}\), which doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students don't understand function notation clearly.

They might think \(\mathrm{h(3)}\) means "3 times h" or get confused about what substitution means, leading them to manipulate the original function incorrectly. This causes them to get stuck and randomly select an answer.

The Bottom Line:

Function evaluation is fundamentally about careful substitution and systematic arithmetic. The key insight is that \(\mathrm{h(3)}\) is simply a shorthand way of asking "what do you get when you replace every x with 3?" Success depends on methodical substitution followed by precise order of operations.