Modulus and Functions Basics
The modulus (or absolute value) always gives you a positive result - think of it as the distance from zero. So |−2| = 2 and |5| = 5. When you see y = |x|, you're looking at a V-shaped graph that never dips below the x-axis.
To solve modulus equations like |3x − 5| = 2 − ½x, you need to consider two cases. First, when the expression inside is positive: 3x − 5 = 2 − ½x, giving x = 2. Second, when it's negative: −(3x − 5) = 2 − ½x, giving x = 6/5.
Functions can be one-to-one (each x-value maps to exactly one y-value, and vice versa) or many-to-one (multiple x-values can give the same y-value). Only one-to-one functions have inverse functions that work properly.
Quick Check: Use the horizontal line test - if any horizontal line crosses the graph more than once, it's many-to-one!
