The exponential function g is defined by \(\mathrm{g(x) = 19 \cdot a^x}\), where a is a positive constant. If \(\mathrm{g(3)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = 19 \cdot a^x}\) (exponential function form)
  • \(\mathrm{g(3) = 2,375}\) (function value at \(\mathrm{x = 3}\))
  • Need to find \(\mathrm{g(4)}\)
• This tells us: When \(\mathrm{x = 3}\), the output is 2,375, so \(\mathrm{19 \cdot a^3 = 2,375}\)

2. INFER the solution strategy

• To find \(\mathrm{g(4)}\), we need to know the value of the base '\(\mathrm{a}\)'
• We can find '\(\mathrm{a}\)' using the given condition \(\mathrm{g(3) = 2,375}\)
• Once we have '\(\mathrm{a}\)', we can calculate \(\mathrm{g(4) = 19 \cdot a^4}\)

3. SIMPLIFY to find the base value '\(\mathrm{a}\)'

• Start with: \(\mathrm{19 \cdot a^3 = 2,375}\)
• Divide both sides by 19: \(\mathrm{a^3 = 2,375 \div 19 = 125}\)
• Take the cube root: \(\mathrm{a = \sqrt[3]{125} = 5}\)

4. SIMPLIFY to calculate \(\mathrm{g(4)}\)

• Now we know \(\mathrm{a = 5}\), so: \(\mathrm{g(4) = 19 \cdot 5^4}\)
• Calculate \(\mathrm{5^4}\): \(\mathrm{5^4 = 625}\)
• Final calculation: \(\mathrm{g(4) = 19 \times 625 = 11,875}\) (use calculator)

Answer: 11,875

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to find \(\mathrm{g(4)}\) directly without first determining the value of '\(\mathrm{a}\)'. They might try to use ratios or patterns between \(\mathrm{g(3)}\) and \(\mathrm{g(4)}\) without recognizing that the base value must be found first. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{19 \cdot a^3 = 2,375}\) but make calculation errors when finding \(\mathrm{a^3 = 125}\) or calculating \(\mathrm{\sqrt[3]{125} = 5}\). Common mistakes include getting \(\mathrm{a^3 = 115}\) instead of 125, or miscalculating \(\mathrm{5^4 = 625}\). These computational errors cascade through to the final answer, leading to incorrect values.

The Bottom Line:

This problem tests whether students understand that exponential functions require finding unknown parameters before making predictions, and whether they can execute multi-step calculations accurately. Success depends on recognizing the logical sequence: use known information → find missing parameter → apply function form.