Piecewise-Defined Functions

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Consider the following function definition:

\[f\left( x \right) = \left\{ \begin{array}{l} - 2x, &  - 1 \le x < 0\\{x^2}, & 0 \le x < 1\end{array} \right.\]

This is an example of a function defined piecewise, which means that the function definition is different on different parts of its domain. For the function above, the domain is \(\left[ { - 1,1} \right]\), but the function definition on \(\left[ { - 1,0} \right)\)is different from the function definition on \(\left[{0,1} \right]\).

The following figure shows the graph of this function:

Piece wise defined functions graph

The range of this function is \(\left[ {0,2} \right]\).

Example 1: Consider the following function:

\[f\left( x \right) = \left\{ \begin{align}&\;\;\;\; 1, & x \in \mathbb{Q}\\ &- 1, & x \notin \mathbb{Q} \end{align} \right.\]

Find the value of \(f\left( 1 \right)\), \(f\left( {\sqrt 2 } \right)\), and \(f\left( \pi  \right)\).

Solution: The function definition says that if the input is a rational number, then the output is 1, else the output is \(- 1\). Thus,

\[\begin{array}{l}f\left( 1 \right) = 1\\f\left( {\sqrt 2 } \right) = f\left(\pi  \right) =  - 1\end{array}\]