The table below shows three values of x and their corresponding values of f(x).x\(\mathrm{f(x)}\)134699Which equation defines \(\mathrm{f(x)}\)?

1. INFER the solution strategy

• Given information: Table showing three x-values and their corresponding f(x) values
• What this tells us: We need to find which equation produces exactly these input-output pairs
• Strategy: Test each answer choice by substituting all three x-values and checking if we get the correct f(x) values

2. SIMPLIFY by testing the first data point \(\mathrm{(x = 1, f(x) = 3)}\)

Substitute \(\mathrm{x = 1}\) into each choice:

• (A) \(\mathrm{f(1) = \sqrt{1} + 2 = 1 + 2 = 3}\) ✓
• (B) \(\mathrm{f(1) = 2\sqrt{1} + 1 = 2(1) + 1 = 3}\) ✓
• (C) \(\mathrm{f(1) = 3\sqrt{1} = 3(1) = 3}\) ✓
• (D) \(\mathrm{f(1) = \sqrt{3 \times 1} = \sqrt{3} \approx 1.73}\) ✗

Eliminate choices D and E since they don't match \(\mathrm{f(1) = 3}\).

3. SIMPLIFY by testing the second data point \(\mathrm{(x = 4, f(x) = 6)}\)

Test remaining choices A, B, and C:

• (A) \(\mathrm{f(4) = \sqrt{4} + 2 = 2 + 2 = 4 \neq 6}\) ✗
• (B) \(\mathrm{f(4) = 2\sqrt{4} + 1 = 2(2) + 1 = 5 \neq 6}\) ✗
• (C) \(\mathrm{f(4) = 3\sqrt{4} = 3(2) = 6}\) ✓

Only choice C remains!

4. SIMPLIFY by verifying with the third data point \(\mathrm{(x = 9, f(x) = 9)}\)

• (C) \(\mathrm{f(9) = 3\sqrt{9} = 3(3) = 9}\) ✓

Perfect match!

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students test only the first data point \(\mathrm{(x = 1)}\) and choose the first equation that works, without testing all given points systematically.

Since choices A, B, and C all give \(\mathrm{f(1) = 3}\), a student might immediately select choice A without further testing. This leads them to select Choice A \(\mathrm{(\sqrt{x} + 2)}\) even though it fails for the other data points.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when computing square roots or applying order of operations.

For example, they might incorrectly calculate \(\mathrm{\sqrt{3 \times 1}}\) as 3 instead of \(\mathrm{\sqrt{3} \approx 1.73}\), or compute \(\mathrm{3 + \sqrt{1}}\) as 3 instead of 4. These calculation errors can lead to wrong eliminations and ultimately selecting an incorrect answer choice.

The Bottom Line:

This problem requires both strategic thinking (testing ALL data points systematically) and careful arithmetic with square roots. Students who rush or test incompletely will likely choose an incorrect answer that works for only some of the given points.