x\(\mathrm{f(x)}\)15313521Some values of the linear function f are shown in the table above. Which of the following defines f?

1. TRANSLATE the problem information

• Given information:
  • Table showing x and f(x) values for a linear function
  • Need to find which equation defines f(x)

2. INFER the approach needed

• Since f is linear, it must have the form \(\mathrm{f(x) = mx + b}\)
• We need to find the slope (m) and y-intercept (b)
• We can use any two points from the table to find the slope

3. SIMPLIFY to find the slope

• Using points \(\mathrm{(1, 5)}\) and \(\mathrm{(3, 13)}\):
• \(\mathrm{m = \frac{13 - 5}{3 - 1}}\)
  \(\mathrm{m = \frac{8}{2}}\)
  \(\mathrm{m = 4}\)

4. SIMPLIFY to find the y-intercept

• Substitute point \(\mathrm{(1, 5)}\) into \(\mathrm{f(x) = mx + b}\):
• \(\mathrm{5 = 4(1) + b}\)
  \(\mathrm{5 = 4 + b}\)
  \(\mathrm{b = 1}\)

5. INFER the complete equation

• \(\mathrm{f(x) = 4x + 1}\)
• This matches Choice C

6. APPLY CONSTRAINTS by verifying

• Check with point \(\mathrm{(5, 21)}\): \(\mathrm{f(5) = 4(5) + 1 = 21}\) ✓

Answer: C. f(x) = 4x + 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope or solving for the y-intercept.

For example, they might calculate the slope as \(\mathrm{\frac{13-5}{1-3} = \frac{8}{-2} = -4}\), mixing up the order of coordinates in the denominator. Using this wrong slope with point \(\mathrm{(1, 5)}\): \(\mathrm{5 = -4(1) + b}\) gives \(\mathrm{b = 9}\), leading to \(\mathrm{f(x) = -4x + 9}\). When they test this against the answer choices, none match exactly, so they end up guessing.

Second Most Common Error:

Poor INFER reasoning about strategy: Students try to work backwards from the answer choices rather than systematically finding m and b.

They might substitute \(\mathrm{x = 1}\) into each choice hoping to get \(\mathrm{f(1) = 5}\), but when multiple choices might work for one point, they don't verify with other points. For instance, they might quickly see that Choice B gives \(\mathrm{f(1) = 3(1) + 2 = 5}\) ✓, think it's correct, and select it without checking the other points.

The Bottom Line:

This problem requires systematic execution of the linear function identification process. Students who rush or skip verification steps often select incorrect answers even when they understand the underlying concepts.