Which table gives three values of x and their corresponding values of h(x) for the given function \(\mathrm{h(x) = x^2...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = x^2 - 3}\)
  • Need to find which table correctly shows \(\mathrm{h(x)}\) values for \(\mathrm{x = 1, 2, and 3}\)
• What this tells us: We need to substitute each x value into the function and calculate the corresponding outputs

2. SIMPLIFY by calculating each function value

For x = 1:

• \(\mathrm{h(1) = (1)^2 - 3}\)
  \(\mathrm{= 1 - 3}\)
  \(\mathrm{= -2}\)

For x = 2:

• \(\mathrm{h(2) = (2)^2 - 3}\)
  \(\mathrm{= 4 - 3}\)
  \(\mathrm{= 1}\)

For x = 3:

• \(\mathrm{h(3) = (3)^2 - 3}\)
  \(\mathrm{= 9 - 3}\)
  \(\mathrm{= 6}\)

3. Match results to answer choices

The correct table should show:

Looking at the choices, only Choice B matches these values exactly.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Order of Operations Confusion: Students might subtract 3 before squaring the x value.

For example, calculating \(\mathrm{h(2)}\) as \(\mathrm{(2 - 3)^2}\)
\(\mathrm{= (-1)^2}\)
\(\mathrm{= 1}\) instead of \(\mathrm{(2)^2 - 3}\)
\(\mathrm{= 4 - 3}\)
\(\mathrm{= 1}\). While this happens to give the correct answer for \(\mathrm{x = 2}\), it produces wrong answers for \(\mathrm{x = 1}\) and \(\mathrm{x = 3}\):

• \(\mathrm{h(1)}\) would become \(\mathrm{(1 - 3)^2 = 4}\) instead of \(\mathrm{-2}\)
• \(\mathrm{h(3)}\) would become \(\mathrm{(3 - 3)^2 = 0}\) instead of \(\mathrm{6}\)

This leads to confusion and guessing since none of the tables match this incorrect pattern.

Second Most Common Error:

Basic Arithmetic Mistakes: Students correctly understand the order of operations but make calculation errors.

For instance, miscalculating \(\mathrm{3^2 - 3 = 6}\) instead of \(\mathrm{9 - 3 = 6}\), or getting the sign wrong when calculating \(\mathrm{1^2 - 3 = -2}\). These errors can lead them to select Choice C or Choice D depending on which calculations they get wrong.

The Bottom Line:

This problem tests whether students can correctly apply order of operations in function evaluation. The key insight is that function notation requires careful attention to the sequence of operations, and even small arithmetic errors can lead to selecting the wrong table.