02.31hybrid

# Hybrid Functions

A hybrid function is formed by defining different rules for the function for different parts of its domain.

#### Example 1

… … $f(x) = \Bigg\{ \begin{matrix} 2x + 1 && x \leqslant 1 \\ -x+2 && x>1 \end{matrix}$

f(x) as shown here is an example of a hybrid function.

Notice in the graph of f(x) the closed and open circles are clearly marked.

Note the format for defining a hybrid function.

• Each section has its own domain.
• A hybrid function consist of two or more such sections joined together into one function.

Recall that to be a function, for any x value, there can be only one y-value. {The vertical line test}

This means in a hybrid function, the different domains can never overlap.

In Example 1 above:

• f(0) = 1
• f(1) = 3
• f(1.1) = 0.9
• f(2) = 0
• etc

## Hybrid Functions on the Calculator

Your CAS calculator can use hybrid functions:

On the Classpad 2,

• the hybrid function icon shown is in the Math3 tab.
• the inequality signs are at the bottom of the first column in the Math3 tab

For a hybrid function, it is sometimes useful to define it, instead of continually having to retype it.

To define any function, we use the define command (ACTION menu, COMMAND submenu)

Enter:
… … define f(x) =

• make sure you use "f" from the ABC tab and "x" from the (boldface) variable list
• then select the hybrid function icon from the Math3 tab
• This will put the empty form shown at the top of the screen on the right.
• Fill in the sections exactly the same as the hybrid function shown above and then press EXE

Now you can use the function f(x) in a variety of ways.

For example

• Substituting values such as f(0) and f(1)

and

• solving for particular values such as f(x) = 0

You can also use your defined function to draw a graph of the hybrid function (see below).

## Graphing hybrid functions on the Calculator

You can produce the graph of a hybrid function on the calculator using the same icon.

Go to the Graphs and Tables page and either

• use a function such as f(x) that you have already defined, or
• enter the hybrid function directly into the screen.

Notice that your calculator does not clearly show the circles at the endpoints of the lines. You must remember to draw them in when you draw your own graph.

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## Hybrid Functions with 3 or more sections

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Hybrid functions are not limited to only having two sections.

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Consider the following example:

… … $f(x) = \Bigg\{ \begin{matrix} 1 && x < -1 \\ x^2 && -1 \leqslant x \leqslant 1 \\ 1 - x && x > 1 \end{matrix}$

This function is continuous over its entire domain because at the endpoints of the different sections, the two rules give the same y-value.

• A graph is continuous if you can draw it without lifting your pen from the paper.
• The gap you see on the screen grab of the calculator is a result of the way it draws graphs. The lines should be joined up.

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To obtain the form for a 3 line hybrid function on your calculator
• Click on the icon then click it again.

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