The function g satisfies \(\mathrm{g(ax - 4) = 3x - 5}\) for all real numbers x, where a is a...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(ax - 4) = 3x - 5}\) for all real numbers x
  • a is a positive constant
  • \(\mathrm{g(7) = 4}\)
  • Need to find: value of a

2. INFER the solution strategy

• The functional equation \(\mathrm{g(ax - 4) = 3x - 5}\) doesn't directly give us \(\mathrm{g(7)}\)
• We need to transform this into standard function form \(\mathrm{g(y) = [expression\: in\: y]}\)
• Key insight: Use substitution to eliminate the x-dependence

3. SIMPLIFY through substitution

• Let \(\mathrm{y = ax - 4}\)
• Solve for x: \(\mathrm{ax = y + 4}\), so \(\mathrm{x = \frac{y + 4}{a}}\)
• Substitute back into \(\mathrm{g(ax - 4) = 3x - 5}\):
  \(\mathrm{g(y) = 3\left(\frac{y + 4}{a}\right) - 5}\)

4. SIMPLIFY the expression

• Distribute: \(\mathrm{g(y) = \frac{3y + 12}{a} - 5}\)
• Separate fractions: \(\mathrm{g(y) = \frac{3y}{a} + \frac{12}{a} - 5}\)
• Now we have \(\mathrm{g(y)}\) in standard form!

5. INFER how to use the given condition

• We know \(\mathrm{g(7) = 4}\), so substitute \(\mathrm{y = 7}\):
• \(\mathrm{g(7) = \frac{3(7)}{a} + \frac{12}{a} - 5 = 4}\)

6. SIMPLIFY to solve for a

• Combine like terms: \(\mathrm{\frac{21}{a} + \frac{12}{a} - 5 = 4}\)
• Simplify: \(\mathrm{\frac{33}{a} - 5 = 4}\)
• Add 5: \(\mathrm{\frac{33}{a} = 9}\)
• Solve for a: \(\mathrm{a = \frac{33}{9} = \frac{11}{3}}\)

Answer: 11/3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to directly use \(\mathrm{g(7) = 4}\) without recognizing they need to transform the functional equation first. They might attempt to substitute \(\mathrm{x = 7}\) into \(\mathrm{g(ax - 4) = 3x - 5}\), getting \(\mathrm{g(7a - 4) = 3(7) - 5 = 16}\). But this gives them \(\mathrm{g(7a - 4) = 16}\), not \(\mathrm{g(7) = 4}\), leading to confusion about how to proceed. This causes them to get stuck and abandon systematic solution, often guessing randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the substitution but make algebraic errors when manipulating fractions. A common mistake is incorrectly distributing \(\mathrm{3\left(\frac{y + 4}{a}\right)}\), writing it as \(\mathrm{\frac{3y}{a} + \frac{4}{a}}\) instead of \(\mathrm{\frac{3y}{a} + \frac{12}{a}}\). This leads to \(\mathrm{g(7) = \frac{21}{a} + \frac{4}{a} - 5 = \frac{25}{a} - 5 = 4}\), giving \(\mathrm{\frac{25}{a} = 9}\), so \(\mathrm{a = \frac{25}{9}}\). Since \(\mathrm{\frac{25}{9}}\) isn't among typical answer choices, this creates confusion and guessing.

The Bottom Line:

This problem requires students to recognize that functional equations often need transformation before direct evaluation. The key insight is using substitution to convert the given form into standard function notation.