x\(\mathrm{g(x)}\)-1250111/2521/625For the exponential function g, the table shows four values of x and their corresponding values of \(\mathrm{g(x)}\...

1. TRANSLATE the problem information

• Given information:
  • Table with 4 coordinate pairs for exponential function \(\mathrm{g(x)}\)
  • Four possible equations to choose from
• What this tells us: I need to find which equation produces all four given outputs

2. INFER the most efficient approach

• Since I have specific answer choices, I can test each one against the table values
• I'll start with the easiest point to check: when \(\mathrm{x = 0}\), \(\mathrm{g(0) = 1}\)
• Any choice that fails this test can be eliminated immediately

3. SIMPLIFY the evaluation for \(\mathrm{x = 0}\)

• Test each choice at \(\mathrm{x = 0}\):
  • Choice A: \(\mathrm{g(0) = -25^0 = -1}\) ✗
  • Choice B: \(\mathrm{g(0) = -(1/25)^0 = -1}\) ✗
  • Choice C: \(\mathrm{g(0) = 25^0 = 1}\) ✓
  • Choice D: \(\mathrm{g(0) = (1/25)^0 = 1}\) ✓
• This eliminates A and B since any number to the power 0 equals 1, but both have negative signs

4. SIMPLIFY the evaluation for \(\mathrm{x = 1}\) using remaining choices

• Test C and D at \(\mathrm{x = 1}\) where \(\mathrm{g(1) = 1/25}\):
  • Choice C: \(\mathrm{g(1) = 25^1 = 25}\) ✗ (way too big)
  • Choice D: \(\mathrm{g(1) = (1/25)^1 = 1/25}\) ✓
• This eliminates C, leaving only D

5. SIMPLIFY verification with remaining points

• Verify D works for \(\mathrm{x = -1}\) where \(\mathrm{g(-1) = 25}\):
  \(\mathrm{g(-1) = (1/25)^{-1}}\)
  \(\mathrm{= 1/(1/25)}\)
  \(\mathrm{= 25}\) ✓
• Verify D works for \(\mathrm{x = 2}\) where \(\mathrm{g(2) = 1/625}\):
  \(\mathrm{g(2) = (1/25)^2 = 1/625}\) ✓

Answer: D. \(\mathrm{g(x) = (1/25)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with negative exponents: Many students struggle with \(\mathrm{(1/25)^{-1}}\) and either avoid checking this point or calculate it incorrectly. They might think \(\mathrm{(1/25)^{-1} = -1/25}\) instead of recognizing that \(\mathrm{b^{-n} = 1/b^n}\), so \(\mathrm{(1/25)^{-1} = 25}\). Without properly verifying this crucial point, they might guess or select the wrong choice based on incomplete checking.

Second Most Common Error:

Poor INFER reasoning about elimination strategy: Some students try to derive the exponential function algebraically using \(\mathrm{g(x) = a(b)^x}\) instead of simply testing the given choices. While this approach works, it's more time-consuming and creates more opportunities for algebraic errors. Students who get bogged down in the algebraic derivation may run out of time or make computational mistakes, leading them to guess among the remaining choices.

The Bottom Line:

This problem rewards systematic testing over complex derivation - the key insight is recognizing that with specific answer choices provided, direct substitution and elimination is more efficient and less error-prone than working from first principles.