The function g is defined by \(\mathrm{g(x) = 150(0.6)^x}\). By what percentage does the value of \(\mathrm{g(x)}\) decrease for every...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 150(0.6)^x}\)
  • Need to find: percentage decrease when x increases by 1

• What this tells us: We need to compare the function value at \(\mathrm{x+1}\) to the value at \(\mathrm{x}\)

2. INFER the mathematical relationship

• To find the percentage change, we need to see how \(\mathrm{g(x+1)}\) relates to \(\mathrm{g(x)}\)
• Since we're dealing with exponential functions, when x increases by 1, we multiply by the base again

3. SIMPLIFY the comparison

• \(\mathrm{g(x) = 150(0.6)^x}\)
• \(\mathrm{g(x+1) = 150(0.6)^{(x+1)} = 150(0.6)^x \times (0.6) = g(x) \times 0.6}\)

• This shows: \(\mathrm{g(x+1) = 0.6 \times g(x)}\)

4. INFER what the factor means

• If \(\mathrm{g(x+1) = 0.6 \times g(x)}\), then the new value is 60% of the original value
• This means 60% of the value remains after each unit increase in x

5. TRANSLATE remaining percentage to decrease percentage

• If 60% remains, then the amount that decreased is: \(\mathrm{100\% - 60\% = 40\%}\)

Answer: B (40)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret what the decay factor 0.6 represents. They might think 0.6 means "decreases by 60%" rather than "60% remains."

Following this logic, they would conclude the percentage decrease is 60%, leading them to select Choice C (60).

Second Most Common Error:

Poor TRANSLATE reasoning: Students struggle to set up the comparison between \(\mathrm{g(x+1)}\) and \(\mathrm{g(x)}\). Instead, they might try to work with specific values or get confused about what "decrease for every increase in x by 1" means mathematically.

This confusion often leads to guessing or selecting Choice D (140) if they somehow add the percentages incorrectly.

The Bottom Line:

This problem tests whether students truly understand exponential decay factors. The key insight is that in exponential decay, the base tells you what fraction remains, not what fraction disappears. Students who memorize "0.6 means 60% decrease" without understanding the underlying concept will consistently get this wrong.