\(\mathrm{g(x) = |x - 2| + 3}\)Which table gives three values of x and their corresponding values of g(x) for...

1. TRANSLATE the function for each x-value

Given: \(\mathrm{g(x) = |x - 2| + 3}\)

We need to find g(x) for x = 0, 2, and 4 by substituting each value into the function.

2. SIMPLIFY each calculation systematically

For x = 0:

• \(\mathrm{g(0) = |0 - 2| + 3}\)
• First compute inside absolute value: \(\mathrm{0 - 2 = -2}\)
• Apply absolute value: \(\mathrm{|-2| = 2}\)
• Add: \(\mathrm{2 + 3 = 5}\)

For x = 2:

• \(\mathrm{g(2) = |2 - 2| + 3}\)
• First compute inside absolute value: \(\mathrm{2 - 2 = 0}\)
• Apply absolute value: \(\mathrm{|0| = 0}\)
• Add: \(\mathrm{0 + 3 = 3}\)

For x = 4:

• \(\mathrm{g(4) = |4 - 2| + 3}\)
• First compute inside absolute value: \(\mathrm{4 - 2 = 2}\)
• Apply absolute value: \(\mathrm{|2| = 2}\)
• Add: \(\mathrm{2 + 3 = 5}\)

3. TRANSLATE results back to table format

The function values are: \(\mathrm{(0,5), (2,3), (4,5)}\), which matches table B.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students confuse the definition of absolute value, thinking that \(\mathrm{|-2| = -2}\) instead of \(\mathrm{|-2| = 2}\).

When they compute \(\mathrm{g(0) = |0 - 2| + 3 = |-2| + 3}\), they incorrectly evaluate \(\mathrm{|-2|}\) as -2, getting \(\mathrm{g(0) = -2 + 3 = 1}\) instead of 5. This may lead them to select Choice A (which shows g(0) = 1).

Second Most Common Error:

Poor order of operations in SIMPLIFY: Students add 3 before taking the absolute value, computing \(\mathrm{|0 - 2 + 3|}\) instead of \(\mathrm{|0 - 2| + 3}\).

For x = 0, they might compute \(\mathrm{|0 - 2 + 3| = |1| = 1}\), getting the wrong value and potentially selecting Choice A.

The Bottom Line:

This problem tests whether students truly understand absolute value as a distance concept (always non-negative) and can maintain proper order of operations when functions contain multiple operations.