Learning Lab

Introduction

Functions which have different rules for each subset of the domain are called hybrid functions.

Sometimes they are referred to as piece-wise defined functions.

An example of a hybrid function is: \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1. \end{cases} \end{align*}\] Note that this hybrid function has three rules, each depending on the value of \(x\) in it’s domain. A hybrid function may have two or more rules.

Example 1

Graph the hybrid function \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1. \end{cases} \end{align*}\]

Solution:

This is a hybrid function with three rules. We consider the graph of each of the rules, noting the restricted domains:

\(\text{Rule 1. }y=-x\) , \(x\leq-1\)

Note that the end point at \(x=-1\) is marked with a filled in circle. This means \(-1\) is in the domain of the function.

Rule 2. \(y=1\) , \(-1<x<1\)

In this case, the open circles indicate that the points \(-1\) and \(1\) are not included in the domain of the function.

Rule 3. \(y=x\) , \(x\geq1\)

The “graphical pieces” from rules \(1\) to \(3\) above can be put together to form the graph of the hybrid function \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1 \end{cases} \end{align*}\]

as shown below.

Example 2

Sketch the graph of \[\begin{align*} y & =f\left(x\right)\\ & =\begin{cases} 1-x, & x<0\\ x^{2}, & x\geq0. \end{cases} \end{align*}\]

Solution:

This function has two rules. First rule is \(f\left(x\right)=1-x\) for \(x<0.\) The second rule is \(f\left(x\right)=x^{2}\) for \(x\geq0.\) Graphing each of these and assembling the “graphical pieces” gives the graph for the hybrid function as shown below:

Note the open circle at \(x=0\) as this is not in the domain of the function \(f\left(x\right)=1-x\). However, \(x=0\) is in the domain of \(f\left(x\right)=x^{2}\) and so is shown with a filled dot.

Exercise

\(1.\ \)Draw a sketch graph of \[\begin{align*} f\left(x\right) & =\begin{cases} x+1, & x<0\\ x-1, & x\geq0. \end{cases} \end{align*}\]

\(2.\ \)Draw a sketch graph of \[\begin{align*} f\left(x\right) & =\begin{cases} x^{2}, & x<0\\ -x^{2}, & x\geq0. \end{cases} \end{align*}\]

\(3.\ \)Draw a sketch graph of \[\begin{align*} f\left(x\right) & =\begin{cases} -1, & x<-2\\ 0, & -2\leq x\leq2\\ 1, & x>2. \end{cases} \end{align*}\]

\(4.\ \)Draw a sketch graph of \[\begin{align*} f(x) & =\begin{cases} x+2, & x<-1\\ 1, & -1\leq x\leq1\\ x, & x>1. \end{cases} \end{align*}\]