Piecewise, sign and modulus properties
Read a piecewise function, plot the sign function, build y = |x − 1| + 2 by moving the V, and read modulus inequalities off a graph.
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Guide
A piecewise function has one rule on each part of its domain. The boundary number belongs to only one rule, so check which part it is in. The constant function f(x) = 3 is a horizontal line; the identity function f(x) = x sends every number to itself.
The sign function gives 1 for positive x, −1 for negative x and 0 at zero. The graph of y = |x − h| + k is the V of y = |x| moved h to the right and k up.
|x − 2| < 3 means x is less than 3 away from 2. On the graph, read where the V is below the line y = 3. |x − 2| > 3 means more than 3 away: the part above the line.