The table above gives the values of the function f for some values of x. Which of the following equations...

1. TRANSLATE the problem information

• Given information:
  • Table of \(\mathrm{x}\) and \(\mathrm{f(x)}\) values
  • Four possible exponential functions
• Need to find: Which function produces these exact values

2. INFER the pattern in the data

• Notice that as \(\mathrm{x}\) increases from 0 to 3, \(\mathrm{f(x)}\) decreases from \(\mathrm{5}\) to \(\frac{5}{8}\)
• This decreasing pattern tells us the exponent must be negative
• Why? With base \(\mathrm{2 \gt 1}\), positive exponents create increasing functions, negative exponents create decreasing functions

3. SIMPLIFY by testing each option systematically

• Start with the easiest point to check: \(\mathrm{x = 0}\)

Testing Option A: \(\mathrm{f(x) = 5(2^{(x + 1)})}\)

• \(\mathrm{f(0) = 5(2^1) = 5(2) = 10}\)
• This doesn't equal 5, so eliminate Option A

Testing Option B: \(\mathrm{f(x) = 5(2^x)}\)

• \(\mathrm{f(0) = 5(2^0) = 5(1) = 5}\) ✓
• \(\mathrm{f(1) = 5(2^1) = 5(2) = 10}\)
• This doesn't equal \(\frac{5}{2}\), so eliminate Option B

Testing Option C: \(\mathrm{f(x) = 5(2^{(-x + 1)})}\)

• \(\mathrm{f(0) = 5(2^1) = 5(2) = 10}\)
• This doesn't equal 5, so eliminate Option C

Testing Option D: \(\mathrm{f(x) = 5(2^{-x})}\)

• \(\mathrm{f(0) = 5(2^0) = 5(1) = 5}\) ✓
• \(\mathrm{f(1) = 5(2^{-1}) = 5(\frac{1}{2}) = \frac{5}{2}}\) ✓
• \(\mathrm{f(2) = 5(2^{-2}) = 5(\frac{1}{4}) = \frac{5}{4}}\) ✓
• \(\mathrm{f(3) = 5(2^{-3}) = 5(\frac{1}{8}) = \frac{5}{8}}\) ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that decreasing function values indicate a negative exponent is needed. Instead, they might randomly test options or choose based on familiar-looking forms.

Without this key insight, they may select Choice B (\(\mathrm{f(x) = 5(2^x)}\)) because it looks like the "standard" exponential function form they remember, even though it produces increasing rather than decreasing values.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when evaluating the exponential expressions, particularly with negative exponents. For example, they might incorrectly calculate \(\mathrm{2^{-1}}\) or confuse the order of operations.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

Success requires both pattern recognition (seeing the decreasing trend) and systematic verification through careful calculation of exponential expressions.