For the function g, \(\mathrm{g(x/k) = 2x + 5}\) for all positive values of x, where k is a positive...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x/k) = 2x + 5}\) for all positive x
  • k is a positive constant
  • \(\mathrm{g(3) = 17}\)
• We need to find the value of k

2. INFER the key relationship

• Since \(\mathrm{g(3) = 17}\), this means when the input to function g equals 3, the output equals 17
• From the functional form \(\mathrm{g(x/k) = 2x + 5}\), we need to find x such that:
  • \(\mathrm{x/k = 3}\) (so the input to g is 3)
  • \(\mathrm{2x + 5 = 17}\) (so the output from g is 17)

3. SIMPLIFY to find x

• From the equation \(\mathrm{2x + 5 = 17}\):
  • \(\mathrm{2x = 17 - 5 = 12}\)
  • \(\mathrm{x = 6}\)

4. SIMPLIFY to find k

• Since \(\mathrm{x/k = 3}\) and \(\mathrm{x = 6}\):
  • \(\mathrm{6/k = 3}\)
  • \(\mathrm{k = 6/3 = 2}\)

5. Verify the solution

• If \(\mathrm{k = 2}\) and \(\mathrm{x = 6}\), then \(\mathrm{x/k = 6/2 = 3}\) ✓
• And \(\mathrm{g(3) = 2(6) + 5 = 17}\) ✓

Answer: B) 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that \(\mathrm{g(3) = 17}\) requires both \(\mathrm{x/k = 3}\) AND \(\mathrm{g(x/k) = 2x + 5 = 17}\) simultaneously.

Instead, they might try to directly substitute \(\mathrm{x = 3}\) into \(\mathrm{g(x/k) = 2x + 5}\), getting \(\mathrm{g(3/k) = 2(3) + 5 = 11}\), then incorrectly set this equal to 17. This leads to confusion about what variable to solve for and may cause them to abandon systematic solution and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the functional notation and think that since \(\mathrm{g(3) = 17}\), they can directly substitute 3 for x in the expression \(\mathrm{2x + 5}\).

This gives them \(\mathrm{2(3) + 5 = 11}\), which doesn't equal 17, leading them to think there's an error in the problem setup. This confusion may lead them to select Choice A (1) by assuming \(\mathrm{k = 1}\) makes the problem "simpler."

The Bottom Line:

This problem requires understanding that functional notation like g(input) means you need to work backwards from both the input value and output value to find the parameter k.