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Hybrid functions

Hybrid functions combine different types of functions. They are useful for modelling real-world scenarios that have different behaviours under different conditions, like tax brackets or shipping rates. Other examples include analysing frictional forces in physics, designing profiles and shapes for vehicles, and modelling free-fall acceleration. Use this resource to learn about hybrid functions.

Functions which have different rules for different intervals of their domain are called hybrid functions. Sometimes, they are called piecewise functions.

An example of a hybrid function is:
\[y=f(x)=\begin{cases} -x & x\leq-1\\
1 & -1<x<1\\
x & x\geq1
\end{cases}\]

This hybrid function has three rules. Each depends on the value of \(x\) in its domain. A function is considered hybrid if it has two or more rules.

Graphs of hybrid functions

To graph a hybrid function, we look at each rule and its specified interval separately. We have to play close attention to the endpoints and correctly use open or solid circles to show them.

Let's see how to graph the example function.

Example 1 – graphing hybrid functions

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

This is a hybrid function with three rules. Let's consider the graph of each of the rules, and pay attention to the restricted domains.

Rule \(1\) states that \(y=-x\) for \(x\leq-1\). Since the less than or equal to sign is used, the endpoint \(-1\) has a solid circle. \(-1\) is included in the domain for this function.

Graph showing a linear graph with formula y equals negative x, restricted to x equal to or less than negative 1

Rule \(2\) states that \(y=1\) for \(-1<x<1\). Since the less than sign is used, the endpoints \(-1\) and \(1\) have an open circle. \(-1\) and \(1\) are not included in the domain for this function.

Graph showing a linear graph with formula y equals 1, restricted to x greater than negative 1 and less than 1

Rule \(3\) states that \(y=x\) for \(x\geq1\). Since the greater than or equal to sign is used, the endpoint \(1\) has a solid circle. \(1\) is included in the domain for this function.

Graph showing a linear graph with formula y equals x, restricted to x equal to or greater than 1

The "graphical pieces" from rules \(1\) to \(3\) can be put together to form the graph of the hybrid function.

Graph showing a hybrid function. For x equal to or less than negative 1, the rule is y equals negative x. For x greater than negative 1 and less than 1, the rule is y equals 1. For x equal to or greater than 1, the rule is y equals x.

Graph the following hybrid function.
\[y = f(x) = \begin{cases} 1-x & x<0\\
x^{2} & x\geq0
\end{cases}\]

This function has two rules. The first rule is >\(f(x)=1-x\) for \(x<0\). Since the less than sign is used, the endpoint \(0\) has an open circle.

The second rule is \(f(x)=x^{2}\) for \(x\geq0\). Since the greater than or equal sign is used, the endpoint \(0\) has a solid circle.

Like in Example 1, we can graph each of these and then assemble the "graphical pieces" to get the graph for the hybrid function.

Graph showing a hybrid function. For x less than 0, the rule is y equals 1 minus x. For x greater than or equal to 0, the rule is y equals x squared.

Exercise – graphing hybrid functions

  1. Graph the following hybrid function.
    \[y = f(x) = \begin{cases} x+1 & x<0\\
    x-1 & x\geq0
    \end{cases}\]
  2. Graph the following hybrid function.
    \[y = f(x) = \begin{cases} x^{2} & x<0\\
    -x^{2} & x\geq0
    \end{cases}\]
  3. Graph the following hybrid function.
    \[y = f(x) = \begin{cases} -1 & x<-2\\
    0 & -2\leq x\leq2\\
    1 & x>2
    \end{cases}\]
  4. Graph the following hybrid function.
    \[y = f(x) = \begin{cases} x+2 & x<-1\\
    1 & -1\leq x\leq1\\
    x & x>1
    \end{cases}\]

  1. \(y = f(x) = \begin{cases} x+1 & x<0\\
    x-1 & x\geq0
    \end{cases}\)
    Graph showing a hybrid function. For x less than 0, the rule is y equals x plus 1. For x greater than or equal to 0, the rule is y equals x-1.
  2. \(y = f(x) = \begin{cases} x^{2} & x<0\\
    -x^{2} & x\geq0
    \end{cases}\)
    Graph showing a hybrid function. For x less than 0, the rule is y equals x squared. For x greater than or equal to 0, the rule is y equals negative x squared.
  3. \(y = f(x) = \begin{cases} -1 & x<-2\\
    0 & -2\leq x\leq2\\
    1 & x>2
    \end{cases}\)
    Graph showing a hybrid function. For x less than negative 2, the rule is y equals negative 1. For x greater than or equal to negative 2 and less than or equal to 2, the rule is y equals 0. For x greater than 2, the rule is y equals 1.
  4. \(y = f(x) = \begin{cases} x+2 & x<-1\\
    1 & -1\leq x\leq1\\
    x & x>1
    \end{cases}\)
    Graph showing a hybrid function. For x less than negative 1, the rule is y equals x plus 2. For x greater than or equal to negative 1 and less than or equal to 1, the rule is y equals 1. For x greater than 1, the rule is y equals x.

Images on this page by RMIT, licensed under CC BY-NC 4.0

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