4.6: Piecewise-defined Functions

Last updated
Save as PDF

Page ID: 40919

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

In certain situations a numerical relationship may follow one pattern of behavior for a while and then exhibit a different kind of behavior. In a situation such as this, it is helpful to use what is known as a piecewise defined function - a function that is defined in pieces.
$clipboard_e5fb448f0dcdee1fda4bfad1acfd51b32.png$
In the above example of a piecewise defined function, we see that the $y$ values for the negative values of $x$ are defined differently than the $y$ values for the positive values of $x$
$clipboard_ea68efb190d5e7d001f6d761a3b94ccf2.png$
Sometimes we are given a graph and need to write a piecewise description of the function it describes.
$clipboard_ed03fecdd1995fec456e9d276b42f4562.png$
The piecewise function pictured above could be described as follows:
$clipboard_e491187c5095500bd1f0a3101eb309f79.png$
Exercises 4.6
Sketch a graph for each of the piecewise functions described below.
$clipboard_e026354e6745991244669d813b61d6e7f.png$
$clipboard_e6799170f4de896a07490ad8c0b53876f.png$
$clipboard_efbd2ed0098530b106cc993fe27cfbd7d.png$
$clipboard_ee0ef787c05c14525afdb48be0573c280.png$
$clipboard_e46b417d8a312b5c6913d6fe0a7df80c9.png$
$clipboard_e200f730d86dd22284bf4859b84a1f477.png$
$clipboard_ef049c647d945aa503559a718b4aa991f.png$
$clipboard_e85126bba75504412431e89cd33abd1e8.png$
$clipboard_e0911629f6e0f21ccd47acc6040501dc0.png$
$clipboard_e58e94e39b1479d445b5cbce2b5e2ba1a.png$
$clipboard_e0eddb56ac46128ccb1fd6e41e7a067d6.png$
$clipboard_ea9c8b6e7ef6b7ae5381e48649f76683f.png$
$clipboard_efb3a2c915ae42a3a5199f03b0bd93ef8.png$
$clipboard_ebf00a711047deecdd86da0371c4f4c7d.png$
$clipboard_eba934cc8a9cc432b1b8ae848e62955d2.png$
$clipboard_e88c7521633e80c00fe26e624660867af.png$
$clipboard_e097d05dc992e398d33e5f317b7b52753.png$
$clipboard_e0da0037a750d774aaeeaa31b3d82d3de.png$