Which of the following function(s) is an accurate description of the graph for the range(s) indicated? i. y = 2x + 4 for -3 <= x <= -1 ii. y = |x| - 1 for -1 <= x <= 2 iii. y = |x + 1| for -1 <= x <= 2 iv. y = (1/2)x - 1 for 2 <= x <= 3 A. i, ii and iii only B. i, ii and iv only C. i and iv only D. ii and iv only

GATE 2018 · General Aptitude · Line Graph · medium

Answer: Formulas i (y = 2x+4 on [-3,-1]), ii (y = |x|-1 on [-1,2]), and iv (y = (1/2)x-1 on [2,3]) accurately describe the graph. Formula iii does not match. The answer is B.

1. Test formula i: y = 2x + 4 on [-3, -1]: At x = -3: y = 2(-3) + 4 = -6 + 4 = -2. At x = -1: y = 2(-1) + 4 = -2 + 4 = 2. So segment i goes from (-3, -2) to (-1, 2) with slope 2. Reading the graph on this interval confirms this straight rising line. Formula i is CORRECT.
2. Test formula ii: y = |x| - 1 on [-1, 2]: At x = -1: y = |-1| - 1 = 1 - 1 = 0. At x = 0: y = |0| - 1 = -1 (vertex, the minimum). At x = 2: y = |2| - 1 = 1. The graph shows a V-shape dipping to -1 at x = 0 on this interval, which exactly matches. Formula ii is CORRECT.
3. Test formula iii: y = |x + 1| on [-1, 2]: At x = -1: y = |-1+1| = 0. At x = 0: y = |0+1| = 1. At x = 2: y = |2+1| = 3. This gives a V-shape with vertex at x = -1 (y=0) rising to the right. The graph does NOT show this pattern on [-1, 2] — the graph shows a V dipping below zero. Formula iii is INCORRECT.
4. Test formula iv: y = (1/2)x - 1 on [2, 3]: At x = 2: y = (1/2)(2) - 1 = 1 - 1 = 0. At x = 3: y = (1/2)(3) - 1 = 1.5 - 1 = 0.5. The graph shows a line segment in this region going from (2, 0) upward with a gentle slope, matching slope 1/2. Formula iv is CORRECT.
5. Identify the correct option: Formulas i, ii, and iv are correct; formula iii is incorrect. This matches option B: i, ii and iv only.