For the function g, if \(\mathrm{g(x + 5) = 2x - 3}\) for all values of x, what is the...

1. TRANSLATE the problem setup

• Given information:
  • \(\mathrm{g(x + 5) = 2x - 3}\) for all values of x
  • Need to find \(\mathrm{g(9)}\)
• What this tells us: The function g is defined by a shifted input relationship

2. INFER the solution strategy

• To find \(\mathrm{g(9)}\), I need to figure out: what value of x makes \(\mathrm{x + 5}\) equal to 9?
• Once I know that x-value, I can substitute it into the right side of the equation: \(\mathrm{2x - 3}\)

3. TRANSLATE to set up the key equation

• Set up: \(\mathrm{x + 5 = 9}\)
• SIMPLIFY: Solve for x: \(\mathrm{x = 9 - 5 = 4}\)

4. SIMPLIFY using substitution

• Now substitute \(\mathrm{x = 4}\) into the expression \(\mathrm{2x - 3}\):
• \(\mathrm{g(9) = 2(4) - 3 = 8 - 3 = 5}\)

Answer: C (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't realize they need to work backwards to find the appropriate x-value. Instead, they might substitute \(\mathrm{x = 9}\) directly into \(\mathrm{2x - 3}\), getting \(\mathrm{2(9) - 3 = 18 - 3 = 15}\).

This may lead them to select Choice E (15).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand which side of the functional equation to use. They might get confused about whether \(\mathrm{g(9)}\) should equal \(\mathrm{2x - 3}\) or think they need to solve \(\mathrm{2x - 3 = 9}\).

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key insight is recognizing that functional equations with shifted inputs require working backwards: to find \(\mathrm{g(9)}\), you must first determine what x-value produces 9 when you add 5 to it.