The functions f and g are defined by the given equations. \(\mathrm{f(x) = 3 + |-2x - x^2|}\) \(\mathrm{g(w) =...
GMAT Advanced Math : (Adv_Math) Questions
The functions f and g are defined by the given equations.
\(\mathrm{f(x) = 3 + |-2x - x^2|}\)
\(\mathrm{g(w) = |\frac{-w}{w-1}| - w + 5}\)
If \(\mathrm{f(-4) = c}\), where \(\mathrm{c}\) is a constant, what is the value of \(\mathrm{g(c)}\)?
1. TRANSLATE the problem requirements
- Given information:
- \(\mathrm{f(x) = 3 + |-2x - x²|}\)
- \(\mathrm{g(w) = |-w/(w-1)| - w + 5}\)
- \(\mathrm{f(-4) = c}\)
- What this tells us: We need to find \(\mathrm{f(-4)}\) first to determine \(\mathrm{c}\), then find \(\mathrm{g(c)}\)
2. SIMPLIFY to find f(-4)
- Substitute -4 for x in \(\mathrm{f(x) = 3 + |-2x - x²|}\):
\(\mathrm{f(-4) = 3 + |-2(-4) - (-4)²|}\)
\(\mathrm{f(-4) = 3 + |8 - 16|}\)
\(\mathrm{f(-4) = 3 + |-8|}\)
\(\mathrm{f(-4) = 3 + 8 = 11}\)
3. INFER the value of c
- Since \(\mathrm{f(-4) = c}\), we know \(\mathrm{c = 11}\)
- Now we need to find \(\mathrm{g(11)}\)
4. SIMPLIFY to find g(11)
- Substitute 11 for w in \(\mathrm{g(w) = |-w/(w-1)| - w + 5}\):
\(\mathrm{g(11) = |-11/(11-1)| - 11 + 5}\)
\(\mathrm{g(11) = |-11/10| - 11 + 5}\)
\(\mathrm{g(11) = 11/10 - 6}\)
\(\mathrm{g(11) = 1.1 - 6 = -4.9}\)
Answer: -4.9 (or -49/10)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Making sign errors when working with negative numbers in the absolute value expression.
Students often calculate \(\mathrm{f(-4) = 3 + |-2(-4) - (-4)²|}\) incorrectly by getting confused with the negative signs. They might compute \(\mathrm{(-4)²}\) as -16 instead of +16, leading to \(\mathrm{f(-4) = 3 + |8 - (-16)| = 3 + |24| = 27}\). This would make \(\mathrm{c = 27}\), leading to \(\mathrm{g(27)}\) which gives a completely different answer and causes confusion.
Second Most Common Error:
Poor order of operations in SIMPLIFY: Not properly handling the absolute value operations or fraction arithmetic.
Students might evaluate the absolute value incorrectly in \(\mathrm{g(11) = |-11/10| - 6}\), perhaps forgetting that \(\mathrm{|-11/10| = 11/10 = 1.1}\), not -1.1. This could lead them to calculate \(\mathrm{g(11) = -1.1 - 6 = -7.1}\), which doesn't match any expected form of the answer.
The Bottom Line:
This problem tests your ability to carefully track negative numbers through multiple function evaluations. Success depends on methodical calculation and proper application of absolute value rules at each step.