Evaluating Piecewise Functions

What is a piecewise function?

A piecewise function uses different formulas on different intervals of $x$. To evaluate it at a specific $x$, you must first determine which piece applies (which interval contains your $x$), then use that piece's formula.

The function

$f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } 0 \leq x < 3 \\ 7 & \text{if } x \geq 3 \end{cases}$

Step-by-step evaluation

Evaluate $f(-2)$

Step 1 — Which piece? $x = -2$. Is $-2 < 0$? Yes → use the first piece: $f(x) = x^2$.

Step 2 — Compute: $f(-2) = (-2)^2 = 4$.

Evaluate $f(1)$

Step 1 — Which piece? $x = 1$. Is $1 < 0$? No. Is $0 \leq 1 < 3$? Yes → use the second piece: $f(x) = 2x + 1$.

Step 2 — Compute: $f(1) = 2(1) + 1 = 3$.

Evaluate $f(5)$

Step 1 — Which piece? $x = 5$. Is $5 < 0$? No. Is $0 \leq 5 < 3$? No. Is $5 \geq 3$? Yes → use the third piece: $f(x) = 7$.

Step 2 — Compute: $f(5) = 7$.

Graphing a piecewise function

Draw each piece only on its interval. Use an open circle (∘) where a piece does NOT include the endpoint, and a closed circle (●) where it does. At the boundaries ($x = 0$ and $x = 3$), check which piece "owns" the endpoint using the $\leq$ vs $<$ signs.

Checking continuity at boundaries

At $x = 0$: Left limit (from piece 1) = $0^2 = 0$. Right value (piece 2) = $2(0) + 1 = 1$. Since $0 \neq 1$, there's a jump discontinuity at $x = 0$.

At $x = 3$: Left limit (piece 2) = $2(3) + 1 = 7$. Right value (piece 3) = $7$. Since $7 = 7$, the function is continuous at $x = 3$.

Try it in the visualization

Drag the $x$-slider across the domain. The graph highlights which piece is active, and the $f(x)$ value updates. Open and closed dots at boundaries show the transition points.