The table shows selected values from function f.Which of the following is the best description of function f?x\(\mathrm{f(x)}\)-116017118219

1. TRANSLATE the problem information

• Given information:
  • Table with x and f(x) values: \((-1,16), (0,17), (1,18), (2,19)\)
  • Need to classify as decreasing/increasing and linear/exponential

2. INFER whether the function is increasing or decreasing

• Look at the pattern as x increases: x goes from \(-1 \rightarrow 0 \rightarrow 1 \rightarrow 2\)
• Observe f(x) values: f(x) goes from \(16 \rightarrow 17 \rightarrow 18 \rightarrow 19\)
• Since f(x) increases as x increases, this is an increasing function

3. INFER the test needed to distinguish linear from exponential

• For linear functions: Check if consecutive differences are constant
• For exponential functions: Check if consecutive ratios are constant
• We need to test differences first since that's simpler

4. SIMPLIFY by calculating consecutive differences

• \(\mathrm{f(0) - f(-1) = 17 - 16 = 1}\)
• \(\mathrm{f(1) - f(0) = 18 - 17 = 1}\)
• \(\mathrm{f(2) - f(1) = 19 - 18 = 1}\)

All differences equal 1, confirming constant rate of change

5. INFER the function type from the pattern

• Constant differences → linear function
• Since differences are positive and function increases → increasing linear

Answer: B. Increasing linear

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the tests for linear vs exponential functions. They might calculate ratios instead of differences, getting \(17/16, 18/17, 19/18\), which are all slightly different values around \(1.06\). Since these ratios aren't exactly equal, they incorrectly conclude it's not exponential, but then they're unsure what it is.

This leads to confusion and guessing between the linear options.

Second Most Common Error:

Conceptual confusion about function behavior: Students might misinterpret what "increasing" means, thinking that because all the f(x) values are positive, the function must be "increasing exponential" since exponential functions grow rapidly.

This may lead them to select Choice D (Increasing exponential) without properly testing the mathematical pattern.

The Bottom Line:

The key insight is knowing that linear functions have constant differences between consecutive outputs while exponential functions have constant ratios. Students who mix up these tests or skip the calculation step entirely will struggle to distinguish between the function types correctly.