Question:The function g is defined by \(\mathrm{g(x) = 3|x - 1/2| + 5/4}\). What is the value of \(\mathrm{g(0)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 3|x - 1/2| + 5/4}\)
• Find: \(\mathrm{g(0)}\)
• This means substitute \(\mathrm{x = 0}\) into the function definition

2. TRANSLATE by substitution

• Replace every x with 0:
  \(\mathrm{g(0) = 3|0 - 1/2| + 5/4}\)
• This becomes: \(\mathrm{g(0) = 3|-1/2| + 5/4}\)

3. SIMPLIFY the absolute value

• Evaluate \(\mathrm{|-1/2|}\):
  • The absolute value of any number is its distance from zero
  • \(\mathrm{|-1/2| = 1/2}\)
• Now we have: \(\mathrm{g(0) = 3(1/2) + 5/4}\)

4. SIMPLIFY through arithmetic

• Multiply: \(\mathrm{3(1/2) = 3/2}\)
• Now we need to add: \(\mathrm{3/2 + 5/4}\)
• Convert to common denominator: \(\mathrm{3/2 = 6/4}\)
• Add: \(\mathrm{6/4 + 5/4 = 11/4}\)

Answer: \(\mathrm{11/4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with absolute value: Students may incorrectly evaluate \(\mathrm{|-1/2|}\) as \(\mathrm{-1/2}\) instead of \(\mathrm{1/2}\), forgetting that absolute value is always non-negative.

If they use \(\mathrm{|-1/2| = -1/2}\), they get:
\(\mathrm{g(0) = 3(-1/2) + 5/4}\)
\(\mathrm{= -3/2 + 5/4}\)
\(\mathrm{= -6/4 + 5/4}\)
\(\mathrm{= -1/4}\)

This leads to confusion since the answer should be positive, causing them to guess or second-guess their work.

Second Most Common Error:

Poor SIMPLIFY execution with fractions: Students may struggle with adding fractions that have different denominators, either forgetting to find a common denominator or making arithmetic errors.

For example, they might incorrectly add \(\mathrm{3/2 + 5/4}\) as \(\mathrm{(3+5)/(2+4) = 8/6}\), leading to a wrong final answer and potential guessing.

The Bottom Line:

This problem tests whether students truly understand that absolute value always produces a non-negative result, and whether they can reliably perform fraction arithmetic. The absolute value concept is the key insight that determines success.