The table shows two values of x and their corresponding values of \(\mathrm{f(x)}\) for a linear function f.x\(\mathrm{f(x)}\)-{3}92-{6}If g is...

1. TRANSLATE the problem information

• Given information:
  • Two points on linear function f: \((-3, 9)\) and \((2, -6)\)
  • g is the inverse function of f
  • Need to find \(\mathrm{g(15)}\)

2. INFER the solution strategy

• To find \(\mathrm{g(15)}\), I need to understand what this means: \(\mathrm{g(15)}\) is the x-value such that \(\mathrm{f(x) = 15}\)
• But first, I need to find the equation of \(\mathrm{f(x)}\) using the two given points

3. SIMPLIFY to find the linear function f(x)

• Calculate the slope: \(\mathrm{m = \frac{-6 - 9}{2 - (-3)} = \frac{-15}{5} = -3}\)
• Use point-slope form with \((-3, 9)\):
  \(\mathrm{f(x) - 9 = -3(x + 3)}\)
  \(\mathrm{f(x) - 9 = -3x - 9}\)
  \(\mathrm{f(x) = -3x}\)

4. INFER how to use the inverse relationship

• Since g is the inverse of f, to find \(\mathrm{g(15)}\) I need to solve: \(\mathrm{f(x) = 15}\)
• This gives me: \(\mathrm{-3x = 15}\)
• Therefore: \(\mathrm{x = -5}\)

Answer: A) -5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what \(\mathrm{g(15)}\) actually means in terms of the inverse function relationship.

Students might think \(\mathrm{g(15)}\) means "substitute 15 into some inverse formula" rather than recognizing it means "find the x-value where \(\mathrm{f(x) = 15}\)." They may try to find an explicit formula for \(\mathrm{g(x)}\) by switching variables, getting confused in the algebraic manipulation, and end up guessing.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when finding the equation of \(\mathrm{f(x)}\).

Students might miscalculate the slope (getting +3 instead of -3) or make sign errors in the point-slope form. If they get \(\mathrm{f(x) = 3x}\) instead of \(\mathrm{f(x) = -3x}\), then solving \(\mathrm{3x = 15}\) gives \(\mathrm{x = 5}\).

This may lead them to select Choice D (5).

The Bottom Line:

The key insight is understanding that inverse functions "undo" each other - \(\mathrm{g(15)}\) asks "what input to f gives output 15?" This conceptual understanding, combined with careful algebra, leads to the correct solution.