The linear function f has slope -{12} and y-intercept b. For a constant k, it is given that \(\mathrm{f^{-1}(k/3) =...

1. TRANSLATE the problem information

• Given information:
  • f is linear with slope -12 and y-intercept b, so \(\mathrm{f(x) = -12x + b}\)
  • \(\mathrm{f^{-1}(k/3) = k - 5}\)
• What this tells us: We have an inverse function relationship to work with

2. TRANSLATE the inverse function condition

• The key insight: If \(\mathrm{f^{-1}(k/3) = k - 5}\), then by definition of inverse functions:
  \(\mathrm{f(k - 5) = k/3}\)
• This converts our inverse function statement into a direct function evaluation

3. INFER the solution strategy

• To find b, we need to:
  • Evaluate \(\mathrm{f(k - 5)}\) using our linear function formula
  • Set this equal to \(\mathrm{k/3}\)
  • Solve the resulting equation for b

4. Evaluate \(\mathrm{f(k - 5)}\)

• \(\mathrm{f(k - 5) = -12(k - 5) + b}\)
• \(\mathrm{f(k - 5) = -12k + 60 + b}\)

5. Set up the equation and SIMPLIFY

• From step 2: \(\mathrm{f(k - 5) = k/3}\)
• So: \(\mathrm{-12k + 60 + b = k/3}\)
• Solving for b: \(\mathrm{b = k/3 + 12k - 60}\)
• SIMPLIFY by combining terms: \(\mathrm{b = k/3 + 36k/3 - 60 = 37k/3 - 60}\)

Answer: C. \(\mathrm{(37/3)k - 60}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert the inverse function condition \(\mathrm{f^{-1}(k/3) = k - 5}\) into the equivalent direct function statement \(\mathrm{f(k - 5) = k/3}\). Instead, they might try to find an explicit formula for \(\mathrm{f^{-1}(x)}\) first, leading to unnecessary complexity and algebraic errors. This confusion causes them to get stuck and abandon systematic solution, leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Even if students set up the equation correctly, they make errors when combining \(\mathrm{k/3 + 12k}\). They might forget to convert \(\mathrm{12k}\) to the same denominator (\(\mathrm{36k/3}\)), getting expressions like \(\mathrm{k/3 + 12k = 13k/3}\) instead of \(\mathrm{37k/3}\). This may lead them to select Choice B (\(\mathrm{(13/3)k - 60}\)).

The Bottom Line:

This problem tests whether students truly understand the reciprocal relationship between functions and their inverses, not just memorization of inverse function formulas. The key breakthrough is realizing that inverse notation can be "flipped" into direct function evaluation.