Question:If \(\mathrm{f(x) = ax + b}\) and \(\mathrm{f(2) = 11}\) and \(\mathrm{f(5) = 20}\), what is the value of \(\mathrm{f^{-1}(29)}\)?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = ax + b}\) (linear function with unknown coefficients)
  • \(\mathrm{f(2) = 11}\) (when \(\mathrm{x = 2}\), the output is 11)
  • \(\mathrm{f(5) = 20}\) (when \(\mathrm{x = 5}\), the output is 20)
• Find: \(\mathrm{f^{-1}(29)}\) (the input value that gives output 29)

2. INFER the approach

• To find \(\mathrm{f^{-1}(29)}\), we first need the explicit form of \(\mathrm{f(x)}\)
• This means finding the values of \(\mathrm{a}\) and \(\mathrm{b}\) using the given conditions
• Once we have \(\mathrm{f(x)}\), we can find its inverse function and evaluate it

3. SIMPLIFY to find the slope (coefficient a)

• Use the slope formula with points (2, 11) and (5, 20):
  • \(\mathrm{a = \frac{20 - 11}{5 - 2}}\)
    \(\mathrm{= \frac{9}{3}}\)
    \(\mathrm{= 3}\)

4. SIMPLIFY to find the y-intercept (coefficient b)

• Substitute into \(\mathrm{f(2) = 11}\):
  • \(\mathrm{11 = 3(2) + b}\)
    \(\mathrm{11 = 6 + b}\)
    \(\mathrm{b = 5}\)
• Therefore: \(\mathrm{f(x) = 3x + 5}\)

5. INFER and SIMPLIFY to find the inverse function

• Start with \(\mathrm{y = 3x + 5}\)
• Solve for \(\mathrm{x}\) in terms of \(\mathrm{y}\):
  • \(\mathrm{y - 5 = 3x}\)
    \(\mathrm{x = \frac{y - 5}{3}}\)
• So \(\mathrm{f^{-1}(x) = \frac{x - 5}{3}}\)

6. SIMPLIFY the final calculation

• \(\mathrm{f^{-1}(29) = \frac{29 - 5}{3}}\)
  \(\mathrm{= \frac{24}{3}}\)
  \(\mathrm{= 8}\)

Answer: D (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about inverse notation: Students might interpret \(\mathrm{f^{-1}(29)}\) as \(\mathrm{\frac{1}{f(29)}}\) instead of the inverse function evaluated at 29.

Following this misconception, they would find \(\mathrm{f(29) = 3(29) + 5 = 92}\), then calculate \(\mathrm{\frac{1}{92} \approx 0.011}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up the system to find \(\mathrm{a}\) and \(\mathrm{b}\), but make arithmetic errors in the calculations or when manipulating the algebra to find the inverse function.

For example, they might incorrectly solve for \(\mathrm{b}\) and get \(\mathrm{f(x) = 3x + 2}\), leading to \(\mathrm{f^{-1}(x) = \frac{x - 2}{3}}\), giving \(\mathrm{f^{-1}(29) = \frac{27}{3} = 9}\). This may lead them to select Choice E (9).

The Bottom Line:

This problem tests whether students truly understand what an inverse function represents (undoing the original function) versus just applying reciprocal operations. The multi-step nature also requires careful algebraic manipulation throughout.