## FG5 Hybrid functions

Functions which have different rules for each subset of the domain are called hybrid functions. Sometimes they are referred to as piecewise defined functions.

### 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*}