\(\mathrm{f(x) = 272(2)^x}\) The function f is defined by the given equation. If \(\mathrm{h(x) = f(x - 4)}\), which of...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{f(x) = 272(2)^x}\)
The function f is defined by the given equation. If \(\mathrm{h(x) = f(x - 4)}\), which of the following equations defines function h?
1. TRANSLATE the function composition
- Given information:
- \(\mathrm{f(x) = 272(2)^x}\)
- \(\mathrm{h(x) = f(x - 4)}\)
- This means: substitute \(\mathrm{(x - 4)}\) wherever we see x in \(\mathrm{f(x)}\)
2. TRANSLATE the substitution
- Replace x with \(\mathrm{(x - 4)}\) in \(\mathrm{f(x)}\):
\(\mathrm{h(x) = f(x - 4) = 272(2)^{(x-4)}}\)
- Now we have \(\mathrm{h(x) = 272(2)^{(x-4)}}\)
3. SIMPLIFY using exponent properties
- Apply the rule: \(\mathrm{a^{(m-n)} = \frac{a^m}{a^n}}\)
- So \(\mathrm{(2)^{(x-4)} = \frac{(2)^x}{(2)^4} = \frac{(2)^x}{16}}\)
- Therefore: \(\mathrm{h(x) = 272 \times \frac{(2)^x}{16}}\)
4. SIMPLIFY the coefficient
- \(\mathrm{h(x) = \frac{272(2)^x}{16} = (272 \div 16)(2)^x}\)
- \(\mathrm{272 \div 16 = 17}\)
- Final result: \(\mathrm{h(x) = 17(2)^x}\)
Answer: A. \(\mathrm{h(x) = 17(2)^x}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students incorrectly apply exponent properties or make calculation errors.
Some students might think \(\mathrm{(2)^{(x-4)} = (2)^x - (2)^4}\), which is completely wrong. Others might correctly get \(\mathrm{\frac{(2)^x}{16}}\) but then make arithmetic errors when dividing 272 by 16, perhaps getting 68 instead of 17.
This may lead them to select Choice B (\(\mathrm{h(x) = 68(2)^x}\)) from calculation errors.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misunderstand what \(\mathrm{f(x - 4)}\) means in terms of substitution.
Instead of substituting \(\mathrm{(x - 4)}\) for x, they might think the transformation affects the base or coefficient in some other way, leading to confusion about how horizontal shifts work in function composition.
This leads to confusion and guessing among the remaining choices.
The Bottom Line:
Function composition with exponential functions requires precise substitution skills and careful application of exponent properties - small errors in either area completely derail the solution.