The table shows two values of x and their corresponding values of \(\mathrm{f(x)}\) for a linear function f.x\(\mathrm{f(x)}\)-392-6If g is...
GMAT Algebra : (Alg) Questions
The table shows two values of \(\mathrm{x}\) and their corresponding values of \(\mathrm{f(x)}\) for a linear function \(\mathrm{f}\).
| \(\mathrm{x}\) | \(\mathrm{f(x)}\) |
|---|---|
| -3 | 9 |
| 2 | -6 |
If \(\mathrm{g}\) is the inverse function of \(\mathrm{f}\), what is the value of \(\mathrm{g(15)}\)?
\(\mathrm{-5}\)
\(\mathrm{-3}\)
\(\mathrm{3}\)
\(\mathrm{5}\)
1. TRANSLATE the problem information
- Given information:
- Two points on linear function f: \((-3, 9)\) and \((2, -6)\)
- g is the inverse function of f
- Need to find \(\mathrm{g(15)}\)
2. INFER the solution strategy
- To find \(\mathrm{g(15)}\), I need to understand what this means: \(\mathrm{g(15)}\) is the x-value such that \(\mathrm{f(x)} = 15\)
- But first, I need to find the equation of \(\mathrm{f(x)}\) using the two given points
3. SIMPLIFY to find the linear function f(x)
- Calculate the slope: \(\mathrm{m} = \frac{-6 - 9}{2 - (-3)} = \frac{-15}{5} = -3\)
- Use point-slope form with \((-3, 9)\):
\(\mathrm{f(x)} - 9 = -3(\mathrm{x} + 3)\)
\(\mathrm{f(x)} - 9 = -3\mathrm{x} - 9\)
\(\mathrm{f(x)} = -3\mathrm{x}\)
4. INFER how to use the inverse relationship
- Since g is the inverse of f, to find \(\mathrm{g(15)}\) I need to solve: \(\mathrm{f(x)} = 15\)
- This gives me: \(-3\mathrm{x} = 15\)
- Therefore: \(\mathrm{x} = -5\)
Answer: A) -5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not understanding what \(\mathrm{g(15)}\) actually means in terms of the inverse function relationship.
Students might think \(\mathrm{g(15)}\) means "substitute 15 into some inverse formula" rather than recognizing it means "find the x-value where \(\mathrm{f(x)} = 15\)." They may try to find an explicit formula for \(\mathrm{g(x)}\) by switching variables, getting confused in the algebraic manipulation, and end up guessing.
This leads to confusion and guessing rather than systematic solution.
Second Most Common Error:
Poor SIMPLIFY execution: Making algebraic errors when finding the equation of \(\mathrm{f(x)}\).
Students might miscalculate the slope (getting +3 instead of -3) or make sign errors in the point-slope form. If they get \(\mathrm{f(x)} = 3\mathrm{x}\) instead of \(\mathrm{f(x)} = -3\mathrm{x}\), then solving \(3\mathrm{x} = 15\) gives \(\mathrm{x} = 5\).
This may lead them to select Choice D (5).
The Bottom Line:
The key insight is understanding that inverse functions "undo" each other - \(\mathrm{g(15)}\) asks "what input to f gives output 15?" This conceptual understanding, combined with careful algebra, leads to the correct solution.
\(\mathrm{-5}\)
\(\mathrm{-3}\)
\(\mathrm{3}\)
\(\mathrm{5}\)