The function g is defined by \(\mathrm{g(x) = x^3 + 1}\). If \(\mathrm{g(m) = 9}\), what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = x^3 + 1}\)
  • Condition: \(\mathrm{g(m) = 9}\)
• This means we need to find the input value m that makes the function output equal 9

2. TRANSLATE the condition into an equation

• Since \(\mathrm{g(m) = 9}\), we substitute m into the function:
  • \(\mathrm{g(m) = m^3 + 1 = 9}\)
• Now we have the equation: \(\mathrm{m^3 + 1 = 9}\)

3. SIMPLIFY to isolate the variable

• Subtract 1 from both sides: \(\mathrm{m^3 = 8}\)
• Take the cube root of both sides: \(\mathrm{m = \sqrt[3]{8} = 2}\)

4. Verify the solution

• Check: \(\mathrm{g(2) = 2^3 + 1 = 8 + 1 = 9}\) ✓

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing cube root with square root operations

Students correctly set up \(\mathrm{m^3 = 8}\), but then think they need to take the square root instead of the cube root. They calculate \(\mathrm{m = \sqrt{8} \approx 2.83}\), which they might round to 3.

This may lead them to select Choice D (3)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding the function evaluation setup

Students might think that \(\mathrm{g(m) = 9}\) means they should substitute 9 for x in the original function, setting up \(\mathrm{g(9) = 9^3 + 1}\) instead of the correct equation \(\mathrm{m^3 + 1 = 9}\).

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can work backwards from a function output to find the input. The key challenge is recognizing that "undoing" the cubic operation requires a cube root, not a square root.