Which one of the given figures P, Q, R, and S represents the graph of the following function? f(x) = |x + 1| - |x - 1| A. P B. Q C. R D. S

GATE 2023 · General Aptitude · Absolute Value · medium

Answer: A. P. The graph is a piecewise linear function: f(x) = -2 for x < -1, f(x) = 2x for -1 <= x <= 1, and f(x) = 2 for x > 1.

1. Region x < -1: both arguments negative: For x < -1, f(x) = -2. Flat line at y = -2.
2. Region -1 <= x <= 1: first argument non-negative, second non-positive: For -1 <= x <= 1, f(x) = 2x. A line through the origin with slope 2. At x = -1: f = -2; at x = 1: f = 2.
3. Region x > 1: both arguments positive: For x > 1, f(x) = 2. Flat line at y = 2.
4. Identify the graph shape and match to options: The graph is a clipped ramp (signum-like) with flat tails at y = -2 and y = 2 and a linear segment of slope 2 in between. This shape is a truncated linear function, also known as a saturating ramp. Among figures P, Q, R, S shown on the next page, graph P shows exactly this: flat at -2 for x < -1, rising linearly to 2 for -1 <= x <= 1, then flat at 2 for x > 1.