Graphing f = {x² if x < 2; 3x - 2 if x ≥ 2}

Algebra 2 · Axiom Academy

EXAMPLE Graphing a Piecewise Function Graph f(x)=x^2 for and f(x)=3x-2 for , then check whether the two pieces meet at the boundary. Graph the piecewise function Determine which formula owns the boundary point x=2 , mark it with the correct type of circle, and check whether the two branches actually meet up. Both pieces evaluate to 4 at x=2 : the parabola's excluded (open) boundary and the line's included (closed) boundary land on the same point, so the graph has no break there. Nice work — you graphed a two-formula function by handling each piece's domain, then checking the boundary carefully. Read each domain condition: it tells you exactly which x-values that piece owns. Evaluate the boundary from both formulas: comparing the two outputs tells you whether the graph will jump or meet up cleanly. Closed vs. open: a closed/filled circle marks a boundary INCLUDED by or ; an open/hollow circle marks one EXCLUDED by or . Check continuity: the left limit, right limit, and function value must all agree. Result: here they do — f(2)=4 from every direction, so this two-formula function is continuous at x=2 . This process — split by domain, compare both formulas at the boundary, then mark circles by inclusion — works for any piecewise function, whether or not it turns out continuous.

This is the written version of the interactive lesson above. See the full Algebra 2 course.