Which of the following is true about the values of 2^(x) and 2x + 2 for x gt 0?

1. TRANSLATE the problem information

• Given: Two functions \(\mathrm{2^x}\) and \(\mathrm{2x + 2}\) for \(\mathrm{x \gt 0}\)
• Question asks: Which statement correctly describes their relationship?

2. INFER the approach

• Since we have multiple choice answers describing different relationships, we need to test specific values to see the pattern
• The key insight: We need to find where (if anywhere) the functions intersect, then determine which is larger on either side

3. SIMPLIFY by testing strategic values

Let's test some values:

At \(\mathrm{x = 1}\):

\(\mathrm{2^1 = 2}\)

\(\mathrm{2(1) + 2 = 4}\)

So \(\mathrm{2^1 \lt 2(1) + 2}\)

At \(\mathrm{x = 2}\):

\(\mathrm{2^2 = 4}\)

\(\mathrm{2(2) + 2 = 6}\)

So \(\mathrm{2^2 \lt 2(2) + 2}\)

At \(\mathrm{x = 3}\):

\(\mathrm{2^3 = 8}\)

\(\mathrm{2(3) + 2 = 8}\)

So \(\mathrm{2^3 = 2(3) + 2}\) (They're equal!)

At \(\mathrm{x = 4}\):

\(\mathrm{2^4 = 16}\)

\(\mathrm{2(4) + 2 = 10}\)

So \(\mathrm{2^4 \gt 2(4) + 2}\)

4. CONSIDER ALL CASES to verify the pattern

• For \(\mathrm{x \lt 3}\): The exponential is smaller than the linear function
• At \(\mathrm{x = 3}\): They're equal
• For \(\mathrm{x \gt 3}\): The exponential is larger than the linear function

This matches Choice C exactly, with \(\mathrm{c = 3}\).

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may test only one or two values instead of systematically finding the crossover point, leading to incorrect conclusions about the overall relationship.

For example, testing only \(\mathrm{x = 1}\) and \(\mathrm{x = 2}\) shows \(\mathrm{2^x \lt 2x + 2}\) in both cases, which might lead students to conclude this is always true. This may lead them to select Choice A (\(\mathrm{2^x \lt 2x + 2}\) for all \(\mathrm{x \gt 0}\)).

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students might find that \(\mathrm{2^x = 2x + 2}\) at \(\mathrm{x = 3}\), but fail to test values on both sides to determine the complete relationship pattern.

Without testing \(\mathrm{x \gt 3}\), they might incorrectly think the functions are only equal at one point without a clear pattern. This leads to confusion and guessing.

The Bottom Line:

This problem requires systematic testing to reveal that exponential functions eventually dominate linear functions, but there's a specific crossover point. Students who don't test enough values miss this fundamental relationship between exponential and linear growth.