\(\mathrm{f(x) = 9(2)^x}\)The function f is defined by the given equation. If \(\mathrm{g(x) = f(3x)}\), which of the following equations...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{f(x) = 9(2)^x}\)
The function f is defined by the given equation. If \(\mathrm{g(x) = f(3x)}\), which of the following equations defines the function g?
- \(\mathrm{27(2)^x}\)
- \(\mathrm{9(8)^x}\)
- \(\mathrm{9(6)^x}\)
- \(\mathrm{81(2)^x}\)
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{f(x) = 9(2)^x}\)
- \(\mathrm{g(x) = f(3x)}\)
- What this tells us: We need to substitute 3x wherever we see x in the function f
2. TRANSLATE the composition notation
- \(\mathrm{g(x) = f(3x)}\) means: Replace every x in f(x) with 3x
- So: \(\mathrm{g(x) = 9(2)^{(3x)}}\)
3. SIMPLIFY using exponent rules
- We have: \(\mathrm{g(x) = 9(2)^{(3x)}}\)
- Apply the power rule: \(\mathrm{a^{(mn)} = (a^m)^n}\)
- So: \(\mathrm{2^{(3x)} = (2^3)^x}\)
- Calculate: \(\mathrm{2^3 = 8}\)
- Therefore: \(\mathrm{g(x) = 9(8)^x}\)
Answer: B) \(\mathrm{9(8)^x}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students don't properly understand function composition notation and try to manipulate the coefficient instead of the exponent.
They might think "3x" means multiply the coefficient by 3, leading to \(\mathrm{g(x) = 27(2)^x}\). This may lead them to select Choice A (\(\mathrm{27(2)^x}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students incorrectly apply exponent rules, perhaps adding instead of using the power rule.
They might compute \(\mathrm{2^{(3x)}}\) as \(\mathrm{(2+3)^x = 6^x}\), giving \(\mathrm{g(x) = 9(6)^x}\). This may lead them to select Choice C (\(\mathrm{9(6)^x}\)).
The Bottom Line:
This problem tests whether students understand function composition (substitution) and can correctly apply exponent rules. The key insight is recognizing that f(3x) requires substituting 3x for x, then using the power rule to simplify the resulting exponent.