Discussion
Which one of the given figures P, Q, R and S represents the graph of the following function?
\[ f(x) = \big| |x + 2| - |x - 1| \big| \]
Answer: A
The graph is P.
Work out \(g(x) = |x + 2| - |x - 1|\) piece by piece:
- For \(x \le -2\): \(g = -(x + 2) + (x - 1) = -3\).
- For \(-2 \le x \le 1\): \(g = (x + 2) + (x - 1) = 2x + 1\).
- For \(x \ge 1\): \(g = (x + 2) - (x - 1) = 3\).
Taking the absolute value, \(f(x) = 3\) for \(x \le -2\) and for \(x \ge 1\), and \(f(x) = |2x + 1|\) in between, which falls to 0 at \(x = -0.5\) and rises back to 3. That is the flat-V-flat shape of P.
Q is \(g(x)\) itself, without the outer absolute value. R and S rise to 9 at \(x = -5\), which would need \(f\) to grow without limit on the left, but \(f\) is constant at 3 there.
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