The function g is defined by \(\mathrm{g(x) = \sqrt{x^2 - 9}}\). What is the value of \(\mathrm{g(5)}\)? 3 4 6...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = \sqrt{x^2 - 9}}\)
  • Need to find: \(\mathrm{g(5)}\)
• What this tells us: We need to substitute \(\mathrm{x = 5}\) into the function

2. SIMPLIFY through substitution and arithmetic

• Substitute \(\mathrm{x = 5}\) into \(\mathrm{g(x) = \sqrt{x^2 - 9}}\):

\(\mathrm{g(5) = \sqrt{5^2 - 9}}\)

• Apply order of operations - exponent first:

\(\mathrm{g(5) = \sqrt{25 - 9}}\)

• Complete the subtraction:

\(\mathrm{g(5) = \sqrt{16}}\)

• Evaluate the square root:

\(\mathrm{g(5) = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill - Order of operations confusion: Students might incorrectly apply order of operations and compute \(\mathrm{(5 - 9)^2}\) instead of \(\mathrm{5^2 - 9}\).

Following this path:

\(\mathrm{g(5) = \sqrt{(5 - 9)^2}}\)

\(\mathrm{= \sqrt{(-4)^2}}\)

\(\mathrm{= \sqrt{16}}\)

\(\mathrm{= 4}\)

Interestingly, this still gives the correct answer by coincidence, but represents flawed mathematical reasoning that would cause errors in similar problems.

Second Most Common Error:

Poor SIMPLIFY execution - Basic arithmetic errors: Students might make computational mistakes like calculating \(\mathrm{5^2 = 10}\) or \(\mathrm{25 - 9 = 14}\).

If they compute \(\mathrm{5^2 = 10}\):

\(\mathrm{g(5) = \sqrt{10 - 9}}\)

\(\mathrm{= \sqrt{1}}\)

\(\mathrm{= 1}\)

(not among the choices, leading to confusion)

If they compute \(\mathrm{25 - 9 = 14}\):

\(\mathrm{g(5) = \sqrt{14} \approx 3.74}\), which might lead them to select Choice (A) (3) as the closest value.

The Bottom Line:

This problem tests fundamental function evaluation skills combined with careful arithmetic. The key is methodically following order of operations while maintaining accuracy in basic computations.