Modulus functions and inequalities

Compare y = f(x), y = |f(x)| and y = f(|x|) for linear, quadratic and cubic functions, then solve an inequality against a straight line by seeing where one graph lies above the other.

Drag on the graph (or focus it and use ← →) to move the tracer. Window: −6 ≤ x ≤ 6.

Function f

Graph to show

Coefficients of f

Inequality with line y = mx + k

Show line and solve

Relation

m (gradient)

k (y-intercept)

Try this

With the linear preset (f(x) = 2x − 3, line y = x + 1), check the shaded solution 0.667 < x < 4.00, then confirm it by solving 2x − 3 = x + 1 and −(2x − 3) = x + 1.
Choose a quadratic and switch between |f(x)| and f(|x|): which part of y = f(x) is reflected in each case, and why is y = f(|x|) always symmetric about the y-axis?
Set m = 0 and make k negative with “graph < line”: explain why |f(x)| < k has no solution, then change to y = f(|x|) and see whether that is still true.

Common error: when solving |f(x)| = g(x) by splitting into the cases f(x) = g(x) and −f(x) = g(x), always check each root on a sketch, because a root from one case may not be a genuine intersection.

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