The function \(\mathrm{f(x)}\) is defined as 19 more than 4 times a number x. If \(\mathrm{y = f(x)}\) is graphed...
GMAT Algebra : (Alg) Questions
The function \(\mathrm{f(x)}\) is defined as 19 more than 4 times a number \(\mathrm{x}\). If \(\mathrm{y = f(x)}\) is graphed in the \(\mathrm{xy}\)-plane, what is the best interpretation of the \(\mathrm{x}\)-intercept?
When \(\mathrm{f(x) = 0}\), the number is \(\mathrm{-\frac{19}{4}}\).
When the number is 0, \(\mathrm{f(x) = 19}\).
The value of \(\mathrm{f(x)}\) increases by 1 for each increase of 4 in the value of the number.
For each increase of 1 in the value of the number, \(\mathrm{f(x)}\) increases by 4.
1. TRANSLATE the problem information
- Given information:
- Function f(x) is defined as '19 more than 4 times a number x'
- Need to interpret the x-intercept of y = f(x)
- TRANSLATE the verbal description to mathematical notation:
\(\mathrm{f(x) = 4x + 19}\)
2. INFER what an x-intercept represents
- The x-intercept is where the graph crosses the x-axis
- This happens when the y-value (or f(x)) equals zero
- So we need to find where \(\mathrm{f(x) = 0}\)
3. SIMPLIFY to find the x-intercept
- Set \(\mathrm{f(x) = 0}\):
\(\mathrm{0 = 4x + 19}\) - Subtract 19 from both sides:
\(\mathrm{-19 = 4x}\) - Divide by 4:
\(\mathrm{x = -19/4}\)
4. INFER the interpretation
- When \(\mathrm{f(x) = 0}\), the number x equals \(\mathrm{-19/4}\)
- This matches exactly with Choice A
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse x-intercept with y-intercept interpretation.
They correctly translate \(\mathrm{f(x) = 4x + 19}\), but then think 'intercept' just means 'substitute something' without understanding what x-intercept specifically means. They might substitute \(\mathrm{x = 0}\) instead of \(\mathrm{f(x) = 0}\), getting \(\mathrm{f(0) = 19}\), and select Choice B (When the number is 0, f(x) = 19).
Second Most Common Error:
Weak INFER skill: Students confuse intercept interpretation with slope interpretation.
They understand that \(\mathrm{f(x) = 4x + 19}\) has slope 4, meaning f(x) increases by 4 for each increase of 1 in x. But they don't recognize that the question asks about x-intercept, not slope. This leads them to select Choice D (For each increase of 1 in the value of the number, f(x) increases by 4).
The Bottom Line:
Success requires clearly distinguishing between x-intercept (where graph crosses x-axis), y-intercept (where graph crosses y-axis), and slope (rate of change). The key insight is that x-intercept means 'when does \(\mathrm{f(x) = 0}\)?'
When \(\mathrm{f(x) = 0}\), the number is \(\mathrm{-\frac{19}{4}}\).
When the number is 0, \(\mathrm{f(x) = 19}\).
The value of \(\mathrm{f(x)}\) increases by 1 for each increase of 4 in the value of the number.
For each increase of 1 in the value of the number, \(\mathrm{f(x)}\) increases by 4.