For x gt 0, the function f is defined as follows: \(\mathrm{f(x)}\) equals 201% of x. Which of the following...

1. TRANSLATE the problem information

• Given information:
  • For \(\mathrm{x \gt 0}\), \(\mathrm{f(x)}\) equals 201% of x
  • Need to identify function type from given choices

• TRANSLATE the percentage: \(\mathrm{201\%}\) of \(\mathrm{x = 201/100 \times x = 2.01x}\)
• So \(\mathrm{f(x) = 2.01x}\) for \(\mathrm{x \gt 0}\)

2. INFER the function type

• Looking at \(\mathrm{f(x) = 2.01x}\), this matches the linear function pattern \(\mathrm{f(x) = mx}\) where:
  • \(\mathrm{m = 2.01}\) (the slope/coefficient)
  • There's no constant term (\(\mathrm{b = 0}\))

• Since the coefficient \(\mathrm{2.01 \gt 0}\), this is an increasing function
• Since it follows \(\mathrm{f(x) = mx}\) (not \(\mathrm{f(x) = a^x}\)), this is linear, not exponential

3. APPLY CONSTRAINTS to select the correct choice

• Eliminate decreasing options (A, B) since coefficient is positive
• Eliminate exponential option (C) since we have \(\mathrm{f(x) = mx}\) form, not \(\mathrm{f(x) = a^x}\) form
• Select increasing linear (D) which matches our analysis

Answer: D. Increasing linear

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about exponential vs linear growth: Students see "201%" and think this must represent exponential growth since it's over 100%. They might reason "anything over 100% means exponential" and select Choice C (Increasing exponential).

However, exponential functions have the form \(\mathrm{f(x) = a^x}\), while this problem gives us \(\mathrm{f(x) = 2.01x}\), which is definitively linear.

Second Most Common Error:

Weak TRANSLATE skill: Students might misinterpret what "201% of x" means, possibly thinking it means the function increases by 201% each time x increases by 1, leading to exponential thinking. This confusion about the mathematical meaning can also lead them to select Choice C (Increasing exponential).

The Bottom Line:

The key insight is recognizing that "201% of x" simply means \(\mathrm{f(x) = 2.01x}\), which is a straightforward linear function with slope 2.01, not an exponential function involving powers of x.