If \(\mathrm{f(x) = 3x - 6}\), and g is the inverse function of f, what is the y-intercept of the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 3x - 6}\)
  • \(\mathrm{g}\) is the inverse function of \(\mathrm{f}\)
  • Need to find y-intercept of \(\mathrm{y = g(x)}\)

• What this tells us: We need to find \(\mathrm{g(x)}\) first, then find where it crosses the y-axis

2. INFER the solution strategy

• To find y-intercept of \(\mathrm{g(x)}\), we need \(\mathrm{g(0)}\)
• But first we must find the inverse function \(\mathrm{g(x)}\)
• Strategy: Find inverse, then evaluate at \(\mathrm{x = 0}\)

3. SIMPLIFY to find the inverse function

• Start with \(\mathrm{f(x) = 3x - 6}\)
• Let \(\mathrm{y = 3x - 6}\)
• Swap x and y: \(\mathrm{x = 3y - 6}\)
• Solve for y: \(\mathrm{x + 6 = 3y}\)
• Therefore: \(\mathrm{y = \frac{x + 6}{3}}\)
• So \(\mathrm{g(x) = \frac{x + 6}{3}}\)

4. SIMPLIFY to find the y-intercept

• Evaluate \(\mathrm{g(0)}\):
  \(\mathrm{g(0) = \frac{0 + 6}{3}}\)
  \(\mathrm{= \frac{6}{3}}\)
  \(\mathrm{= 2}\)
• The y-intercept is \(\mathrm{(0, 2)}\)

Answer: B. \(\mathrm{(0, 2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the y-intercept of \(\mathrm{f(x)}\) instead of \(\mathrm{g(x)}\)

They see \(\mathrm{f(x) = 3x - 6}\) and immediately calculate \(\mathrm{f(0) = 3(0) - 6 = -6}\), thinking the answer is \(\mathrm{(0, -6)}\). They miss the crucial step that the question asks for the y-intercept of the inverse function, not the original function.

This leads them to select Choice D. \(\mathrm{(0, -6)}\)

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when finding the inverse

Students start correctly by trying to find the inverse, but make sign errors or fraction mistakes. For example, they might get \(\mathrm{g(x) = \frac{x - 6}{3}}\) instead of \(\mathrm{g(x) = \frac{x + 6}{3}}\), leading to \(\mathrm{g(0) = \frac{-6}{3} = -2}\).

This may lead them to select Choice A. \(\mathrm{(0, -2)}\)

The Bottom Line:

This problem requires students to understand that finding properties of an inverse function requires actually finding the inverse first - you can't just work with the original function.