For the linear function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane passes through the points \(\mathrm{(2, 9)}\)...

1. TRANSLATE the problem information

• Given information:
  • Linear function f passes through points (2, 9) and (6, 17)
  • Need to find \(\mathrm{f^{-1}(x)}\)
• What this tells us: We need to find the original function \(\mathrm{f(x)}\) first, then find its inverse

2. INFER the approach

• To find an inverse function, we must know the original function
• Strategy: Find \(\mathrm{f(x)}\) using the two points, then apply inverse process
• We'll use slope formula and point-slope form to find \(\mathrm{f(x)}\)

3. SIMPLIFY to find the slope of \(\mathrm{f(x)}\)

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{17 - 9}{6 - 2}}\)
  \(\mathrm{= \frac{8}{4}}\)
  \(\mathrm{= 2}\)

4. SIMPLIFY to find the equation of \(\mathrm{f(x)}\)

• Using point-slope form with (2, 9): \(\mathrm{y - 9 = 2(x - 2)}\)
• Expand: \(\mathrm{y - 9 = 2x - 4}\)
• Solve for y: \(\mathrm{y = 2x + 5}\)
• Therefore: \(\mathrm{f(x) = 2x + 5}\)

5. INFER the inverse process

• To find \(\mathrm{f^{-1}(x)}\), we swap x and y in the equation, then solve for y
• This reverses the input-output relationship

6. SIMPLIFY to find \(\mathrm{f^{-1}(x)}\)

• Start with: \(\mathrm{y = 2x + 5}\)
• Swap variables: \(\mathrm{x = 2y + 5}\)
• Solve for y: \(\mathrm{x - 5 = 2y}\)
• Divide by 2: \(\mathrm{y = \frac{x - 5}{2}}\)
  \(\mathrm{= \frac{1}{2}x - \frac{5}{2}}\)
• Therefore: \(\mathrm{f^{-1}(x) = \frac{1}{2}x - \frac{5}{2}}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to find the inverse without first determining \(\mathrm{f(x)}\). They might try to work directly from the coordinate points to find \(\mathrm{f^{-1}(x)}\), not realizing they need the equation of the original function first.

This leads to confusion about how to proceed and often results in guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students find \(\mathrm{f(x) = 2x + 5}\) correctly but make algebraic errors when finding the inverse. Common mistakes include:

• Forgetting to swap x and y variables
• Incorrectly distributing when solving \(\mathrm{x = 2y + 5}\)
• Sign errors when rearranging to solve for y

This may lead them to select Choice B (\(\mathrm{\frac{1}{2}x + \frac{5}{2}}\)) or other incorrect options.

The Bottom Line:

This problem requires students to recognize that finding an inverse is a two-step process: first find the original function, then apply the inverse procedure. The algebraic manipulation skills must be solid in both steps.