When more than one function is needed to create a model for a given situation, we can use a **piecewise function** to connect the different pieces. A domain is given for each individual function which makes up the piecewise function.

To graph piecewise functions, it can be helpful to first find all of the endpoints, then determine what the graph will look like between them.

The inequalities for the domains help us determine whether the endpoints are filled (closed) or unfilled (open).

Filled (closed): \bullet for \leq or \geq\\Unfilled (open): \circ for < or >

The domain of the function is all real numbers except for 0 because 0 is the only value where there is no solid graph.

The range is all real numbers except for those between -4 and -5. -4 is included in the domain because there is solid graph at a y-value of -4 but -5 is not included in the range.

A **discontinuity** occurs when there are values missing in the domain or range. Graphically, a discontinuity appears as a hole or a gap in the graph. For example, the graph above has a discontinuity at x=-2 and at x=0.

Consider the piecewise function: f\left(x\right)=\begin{cases} x + 6, & x \lt -4 \\ 4, &-4\leq x \lt -2 \\ \left(x+2\right)^{2} +5, & x \gt -2 \end{cases}

a

Evaluate the function at f\left(-7\right),\,f\left(-4\right),\,f\left(-2\right),\,f\left(1\right).

Worked Solution

b

Graph the piecewise function.

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c

Describe the key features of the function graphed in part (b).

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Write the piecewise-defined function based on the graph shown:

Worked Solution

Determine the types of functions that are in this piecewise function.

f\left(x\right) = \begin{cases} x+4, & x \lt 0 \\ -2, & 0 \leq x \lt 4 \\ 12-x^2, & x\geq 4 \end{cases}

Worked Solution

Consider the functions shown:

- Function 1: f\left(x\right) = \begin{cases} -x^{2}+5, & x \leq 0 \\ -\dfrac{1}{2}x-3, & 0<x \end{cases}
- Function 2:

a

Determine which function is increasing if x>0.

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b

Determine whether or not each function has a maximum or minimum value.

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Idea summary

A piecewise function is made up of two or more functions.

The domain of a piecewise function is visible in both the piecewise-defined function and the graph.