2.1 Piecewise Functions - Introduction to Calculus

Piecewise Functions

Piecewise functions typically feature one or more points at which the function changes from one form to another. To graph a piecewise function, simply graph each piece and then restrict it to its designated domain. Pay special attention when plotting the breaking point (closed circle includes the point, open circle excludes the point).

Move the input across a boundary. Only the rule whose domain includes that input supplies the value.

Function

\displaystyle f(x)=\begin{cases}1 & x\le0\\[.4em]4^x & x>0\end{cases}

\displaystyle y=1,\quad x\le0

\displaystyle y=4^x,\quad x>0

Input x0

Constant branch: x ≤ 0

\displaystyle f(0)=1

Filled circles include an endpoint; hollow circles exclude it. In the first function, both pieces meet at (0, 1), and the constant piece includes that point.

QUESTION

Which marker includes a boundary point in a graph?

Graph the piecewise function given by

\displaystyle y=\begin{cases}1 & x\le0\\[.6em]4^x & x>0\end{cases}

Graph the piecewise function given by

\displaystyle y=\begin{cases}\dfrac12x+3 & x<-2\\[.6em]0 & -2\le x\le2\\[.6em]x^2-1 & x>2\end{cases}

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Piecewise Functions

Move the input across a boundary. Only the rule whose domain includes that input supplies the value.

Function

\displaystyle f(x)=\begin{cases}1 & x\le0\\[.4em]4^x & x>0\end{cases}

\displaystyle y=1,\quad x\le0

\displaystyle y=4^x,\quad x>0

Input x0

Constant branch: x ≤ 0

\displaystyle f(0)=1

Filled circles include an endpoint; hollow circles exclude it. In the first function, both pieces meet at (0, 1), and the constant piece includes that point.

QUESTION

Which marker includes a boundary point in a graph?

Graph the piecewise function given by

\displaystyle y=\begin{cases}1 & x\le0\\[.6em]4^x & x>0\end{cases}

Graph the piecewise function given by

\displaystyle y=\begin{cases}\dfrac12x+3 & x<-2\\[.6em]0 & -2\le x\le2\\[.6em]x^2-1 & x>2\end{cases}

Piecewise Functions

Piecewise Functions

Follow a piecewise function

KNOWLEDGE QUIZ

Graph a two-part function

Graph a three-part function

Piecewise Functions

Piecewise Functions

Follow a piecewise function

KNOWLEDGE QUIZ

Graph a two-part function

Graph a three-part function