The functions g and h are defined by \(\mathrm{g(x) = 3x - 2}\) and \(\mathrm{h(x) = x^2 + 1}\). What...

1. TRANSLATE the composition notation

• Given information:
  • \(\mathrm{g(x) = 3x - 2}\)
  • \(\mathrm{h(x) = x^2 + 1}\)
  • Find \(\mathrm{g(h(3))}\)
• What this notation means: Evaluate \(\mathrm{h(3)}\) first, then use that result as the input for function g

2. INFER the correct approach

• Function composition works from inside-out
• Step 1: Find \(\mathrm{h(3)}\)
• Step 2: Use \(\mathrm{h(3)}\) as the input for g

3. SIMPLIFY by evaluating the inner function first

• Calculate \(\mathrm{h(3)}\):
  \(\mathrm{h(3) = 3^2 + 1}\)
  \(\mathrm{h(3) = 9 + 1}\)
  \(\mathrm{h(3) = 10}\)

4. SIMPLIFY by evaluating the outer function

• Now calculate \(\mathrm{g(10)}\):
  \(\mathrm{g(h(3)) = g(10)}\)
  \(\mathrm{g(10) = 3(10) - 2}\)
  \(\mathrm{g(10) = 30 - 2}\)
  \(\mathrm{g(10) = 28}\)

Answer: 28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret \(\mathrm{g(h(3))}\) as requiring separate evaluation of \(\mathrm{g(3)}\) and \(\mathrm{h(3)}\), then trying to combine these results.

They calculate \(\mathrm{g(3) = 3(3) - 2 = 7}\) and \(\mathrm{h(3) = 10}\), then attempt operations like \(\mathrm{7 + 10 = 17}\) or \(\mathrm{7 \times 10 = 70}\), neither of which appears in the answer choices. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students understand they need function composition but reverse the order, attempting to evaluate g first, then h.

They might try \(\mathrm{g(3) = 7}\), then \(\mathrm{h(7) = 7^2 + 1 = 50}\), leading them to select an answer not among the choices, causing them to get stuck and guess.

The Bottom Line:

Function composition notation can be tricky because it reads differently than the order of operations. The key insight is recognizing that \(\mathrm{g(h(3))}\) means "start with the innermost function and work outward," just like parentheses in regular arithmetic.