The table shows selected values of a function h, where \(\mathrm{h(x) = \frac{x - 2}{f(x)}}\) and f is a linear...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(x) = \frac{x - 2}{f(x)}}\) where f is linear
  • Table with three \(\mathrm{(x, h(x))}\) pairs
• We need: y-intercept of \(\mathrm{f(x)}\)

2. INFER the approach

• Since we have \(\mathrm{h(x)}\) but need information about \(\mathrm{f(x)}\), we should rearrange the given relationship
• From \(\mathrm{h(x) = \frac{x - 2}{f(x)}}\), we can solve for \(\mathrm{f(x)}\): \(\mathrm{f(x) = \frac{x - 2}{h(x)}}\)
• This will give us actual points on the linear function \(\mathrm{f(x)}\)

3. SIMPLIFY to find f(x) values

• At \(\mathrm{x = -5}\): \(\mathrm{f(-5) = \frac{-5 - 2}{7/9}}\)
  \(\mathrm{= -7 \div \frac{7}{9}}\)
  \(\mathrm{= -7 \times \frac{9}{7}}\)
  \(\mathrm{= -9}\)
• At \(\mathrm{x = 1}\): \(\mathrm{f(1) = \frac{1 - 2}{-1/9}}\)
  \(\mathrm{= -1 \div \frac{-1}{9}}\)
  \(\mathrm{= -1 \times (-9)}\)
  \(\mathrm{= 9}\)
• At \(\mathrm{x = 4}\): \(\mathrm{f(4) = \frac{4 - 2}{1/9}}\)
  \(\mathrm{= 2 \div \frac{1}{9}}\)
  \(\mathrm{= 2 \times 9}\)
  \(\mathrm{= 18}\)

4. INFER how to use the linear nature

• Since f is linear, it has the form \(\mathrm{f(x) = mx + b}\)
• We can use any two points to find the slope m, then find b

5. SIMPLIFY to find slope and y-intercept

• Using points \(\mathrm{(1, 9)}\) and \(\mathrm{(4, 18)}\):
• Slope: \(\mathrm{m = \frac{18 - 9}{4 - 1}}\)
  \(\mathrm{= \frac{9}{3}}\)
  \(\mathrm{= 3}\)
• Using point \(\mathrm{(1, 9)}\): \(\mathrm{9 = 3(1) + b}\)
  \(\mathrm{b = 6}\)

Answer: C. (0, 6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see \(\mathrm{h(x) = \frac{x - 2}{f(x)}}\) but don't recognize they need to rearrange this to find \(\mathrm{f(x)}\) values. Instead, they might try to work directly with the \(\mathrm{h(x)}\) values or attempt to substitute into a general linear form without first finding actual points on \(\mathrm{f(x)}\). This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{f(x) = \frac{x - 2}{h(x)}}\) but make errors when dividing by fractions. For example, calculating \(\mathrm{f(-5) = \frac{-7}{7/9}}\) incorrectly as \(\mathrm{(-7) \times \frac{7}{9}}\) instead of \(\mathrm{(-7) \times \frac{9}{7}}\). This gives wrong \(\mathrm{f(x)}\) values, leading to incorrect slope and y-intercept calculations. This may lead them to select Choice A (0, -6) or Choice D (0, 9).

The Bottom Line:

This problem requires students to think backwards from the composed function \(\mathrm{h(x)}\) to recover the original linear function \(\mathrm{f(x)}\), then apply standard linear function techniques. The key insight is recognizing the algebraic rearrangement needed to extract useful information.