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Which one of the given figures P, Q, R and S represents the graph of the following function?

\[ f(x) = \big|\,|x+2| - |x-1|\,\big| \]

  • A.Graph P of f(x): equal to 3 up to x = -2, a V down to 0 at x = -0.5, back to 3 at x = 1, then constant at 3
  • B.Graph Q of f(x): equal to -3 up to x = -2, rising linearly to 3 at x = 1, then constant at 3
  • C.Graph R of f(x): falling from 9 at x = -5 to 0 at x = -0.5, rising to 3 at x = 1, then constant at 3
  • D.Graph S of f(x): falling from 9 at x = -5 to 3 at x = -2, then constant at 3

Answer: A

The graph is P.

Work out \(g(x) = |x+2| - |x-1|\) on the three intervals set by x = -2 and x = 1:

\[ \begin{aligned} x \le -2:&\quad g = -(x+2)+(x-1) = -3 \\ -2 \le x \le 1:&\quad g = (x+2)+(x-1) = 2x+1 \\ x \ge 1:&\quad g = (x+2)-(x-1) = 3 \end{aligned} \]

Taking the absolute value, f is 3 for x up to -2, falls along |2x + 1| to 0 at x = -1/2, rises back to 3 at x = 1, and stays at 3. That is graph P.

Q is the graph of g itself, without the outer absolute value. R and S rise without bound to the left, which f cannot do, since it never exceeds 3.

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