Piecewise Functions

What is a piecewise function?

A piecewise function uses different formulas on different intervals of $x$. To evaluate it at a specific $x$, first determine which interval $x$ falls in, then use that piece's formula.

Example: $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x+1 & \text{if } x \geq 0 \end{cases}$

At $x = -3$: use $x^2$ → $f(-3) = 9$. At $x = 2$: use $2x+1$ → $f(2) = 5$.

Checking continuity at the boundary

At $x = 0$: left limit = $0^2 = 0$; right value = $2(0)+1 = 1$. Since $0 \neq 1$, there's a jump discontinuity.

Graphing rules

• Draw each piece only on its interval.
• Use open circle (∘) at endpoints not included ($<$ or $>$).
• Use closed circle (●) at endpoints included ($\leq$ or $\geq$).

Try it in the visualization

Toggle each piece on/off. Open/closed dots at boundaries show ownership. Adjust the second piece's slope and intercept to make the function continuous (dots meet). Continuityrequires the left and right limits to agree at each boundary. Here at $x=0$: left limit = $0^2 = 0$, right value = $2(0)+1 = 1$. Gap of 1 → discontinuous.