Piecewise Functions Reference Cheat Sheet

Piecewise functions use different rules for different parts of the domain. Students need this cheat sheet to read interval conditions, choose the correct formula, and evaluate values accurately. It is especially useful for graphing real-world models such as tax brackets, shipping costs, and step pricing.

The goal is to make each piece of the function clear and organized.

The most important idea is that the input value decides which rule to use. Each condition, such as $x < 0$ or $x \ge 3$ , tells where that formula applies. Graphing a piecewise function means graphing each rule only on its assigned interval.

Open and closed circles show whether endpoint values are included.

Key Facts

A piecewise function is written with multiple rules, such as $f(x)=\begin{cases}x+2, & x<1 \\ 3x, & x\ge 1\end{cases}$ .
To evaluate $f(a)$ , first find which condition contains $x=a$ , then substitute $a$ into only that rule.
A condition with $<$ or $>$ uses an open circle at the endpoint because the endpoint is not included.
A condition with $\le$ or $\ge$ uses a closed circle at the endpoint because the endpoint is included.
The domain is the set of all allowed input values $x$ covered by the conditions.
The range is the set of all output values $y$ produced by the pieces of the function.
A piecewise function is continuous at $x=a$ when the left-hand value, right-hand value, and function value are all equal: $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$ .
If two conditions overlap, such as $x\le 2$ and $x\ge 2$ , the function may be undefined or ambiguous at $x=2$ unless the rules give the same output.

Vocabulary

Piecewise function: A function defined by different formulas on different parts of its domain.
Interval: A set of input values between given endpoints, such as $-2\le x<5$ .
Endpoint: A boundary value where one piece of a piecewise function starts or stops.
Open circle: A graph symbol showing that an endpoint is not included, usually for $<$ or $>$ .
Closed circle: A graph symbol showing that an endpoint is included, usually for $\le$ or $\ge$ .
Continuity: A property where a graph has no break, jump, or hole at a point.

Common Mistakes to Avoid

Using every formula to evaluate one input is wrong because only the condition containing that input should be used.
Ignoring endpoint symbols is wrong because $x<3$ and $x\le 3$ give different inclusion rules at $x=3$ .
Graphing each rule across all real numbers is wrong because each formula only applies on its stated interval.
Choosing the wrong piece for negative numbers is wrong because inequalities must be checked carefully, especially with conditions like $x<-1$ and $x\ge -1$ .
Assuming a piecewise graph is always continuous is wrong because jumps, holes, or mismatched endpoint values can occur.

Practice Questions

1 For $f(x)=\begin{cases}2x+1, & x<3 \\ x^2, & x\ge 3\end{cases}$ , find $f(2)$ and $f(3)$ .
2 For $g(x)=\begin{cases}-x, & x\le 0 \\ x+4, & x>0\end{cases}$ , find $g(-5)$ , $g(0)$ , and $g(2)$ .
3 Graph $h(x)=\begin{cases}1, & x<-2 \\ x+3, & -2\le x<1 \\ 4, & x\ge 1\end{cases}$ and label all open and closed circles.
4 A piecewise function has $f(x)=x+2$ for $x<1$ and $f(x)=5$ for $x\ge 1$ . Explain whether the function is continuous at $x=1$ .

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Piecewise Functions Reference Cheat Sheet

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