Piecewise Functions

Calculus Readiness · Axiom Academy

One function, several rules — each ruling its own stretch of the number line, stitched together at the breakpoints. 1. One Function, Different Rules A piecewise function is defined by different expressions on different parts of its domain. It's written with a single tall brace — each row pairs a rule with the interval where that rule applies: Read it row by row: this rule, on this interval. Here's a concrete one. It has two breakpoints , at x = 0 and x = 3 , where the active rule changes: Watch the input slide across the number line below. As it crosses each breakpoint, the active rule lights up and the output dot jumps onto that piece's graph — the function is literally switching formulas as you cross over. 2. Evaluating: Find the Interval, Then Plug In To evaluate f(a) , you never use all the rules at once. You do three things: Check which interval the input a falls into. Use the rule for that interval. Substitute into that rule only — ignore the others. The animation drops a marker on each input, lights the interval that contains it, and fires only the matching rule. Same function as Step 1. -2 < 0 , so the top rule: 2(-2)+1 = -3 . 0 is in [0,3] , so the middle rule: 0^2 = 0 . 2 is in [0,3] , middle rule: 2^2 = 4 . 5 > 3 , so the bottom rule: f(5) = 9 . Watch the endpoint inequalities

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