The table shows three values of x and their corresponding values of \(\mathrm{g(x)}\), where \(\mathrm{g(x)} = \frac{\mathrm{f(x)}}{\mathrm{x}^2 + 1}\...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = \frac{f(x)}{x^2 + 1}}\) where \(\mathrm{f}\) is linear
  • Three coordinate pairs: \(\mathrm{(-1, 0), (0, 10), (2, 6)}\)

• Since \(\mathrm{f}\) is linear, we can write \(\mathrm{f(x) = mx + b}\)
• This means \(\mathrm{g(x) = \frac{mx + b}{x^2 + 1}}\)

2. TRANSLATE each table entry into an equation

• For \(\mathrm{x = -1}\), \(\mathrm{g(-1) = 0}\):
  \(\mathrm{\frac{m(-1) + b}{(-1)^2 + 1} = 0}\)
  \(\mathrm{\frac{-m + b}{2} = 0}\)

• For \(\mathrm{x = 0}\), \(\mathrm{g(0) = 10}\):
  \(\mathrm{\frac{m(0) + b}{0^2 + 1} = 10}\)
  \(\mathrm{\frac{b}{1} = 10}\)

• For \(\mathrm{x = 2}\), \(\mathrm{g(2) = 6}\):
  \(\mathrm{\frac{m(2) + b}{2^2 + 1} = 6}\)
  \(\mathrm{\frac{2m + b}{5} = 6}\)

3. SIMPLIFY each equation

• From the first equation: \(\mathrm{-m + b = 0}\), so \(\mathrm{m = b}\)
• From the second equation: \(\mathrm{b = 10}\)
• From the third equation: \(\mathrm{2m + b = 30}\)

4. INFER the solution strategy and solve

• We have \(\mathrm{m = b}\) and \(\mathrm{b = 10}\), so \(\mathrm{m = 10}\)
• Check with the third equation: \(\mathrm{2(10) + 10 = 30}\) ✓
• Therefore \(\mathrm{f(x) = 10x + 10}\)

5. INFER the final answer

• The y-intercept occurs when \(\mathrm{x = 0}\)
• \(\mathrm{f(0) = 10(0) + 10 = 10}\)
• The y-intercept is \(\mathrm{(0, 10)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not correctly convert the rational function relationship into proper equations, especially struggling with the denominator \(\mathrm{x^2 + 1}\).

For example, they might incorrectly write \(\mathrm{g(-1) = \frac{-m + b}{-1 + 1} = \frac{-m + b}{0}}\), creating undefined expressions, or forget to square the x-value in the denominator. This leads to confusion and incorrect equation setup, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving the system.

A common mistake is incorrectly solving \(\mathrm{-m + b = 0}\). Some students might think this gives \(\mathrm{b = m}\) instead of \(\mathrm{m = b}\), leading to \(\mathrm{b = 10}\) and \(\mathrm{m = 10}\), but then getting confused about which is which. Others might make arithmetic errors when verifying with the third equation. This may lead them to select Choice D (0, 30) by incorrectly using the value 30 that appears in their work.

The Bottom Line:

This problem requires careful translation of multiple conditions into a consistent algebraic system. The combination of rational functions and linear relationships creates multiple opportunities for both setup errors and computational mistakes.