Question 5

A function $$f \colon \mathbb{R} \rightarrow \mathbb{R}$$ is defined by $$f(x) = |x - 1| + |x - 2| + |x - 3|$$ for all real numbers $$x$$. Then

Solution

For $$x < 1$$ all three expressions inside the moduli are negative, so $$f(x) = (1 - x) + (2 - x) + (3 - x) = 6 - 3x$$, which holds in particular for all $$x < 0$$. For $$x > 3$$, $$f(x) = 3x - 6$$, so the third and fourth options fail, and the second fails since $$f(0) = 6 \ne 0$$.

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