Piecewise Functions

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Piecewise Function Definitions

Definition: a function with different definitions for different non-overlapping parts of its domain, together covering the whole domain.

Piecewise function definitions usually translate very easily into programming language “if” statements.

Example: absolute value

• a(x) = x if x ≥ 0
• a(x) = -x otherwise

Try coding this in Matlab.

function [ absX ] = absoluteValue( x )
    %ABSOLUTEVALUE. Find the absolute value of x.
    if x >= 0
        absX = x
    else
        absX = -x
    end
end

Notice how each “arm” of the “if” is a straightforward translation into Matlab of the corresponding case in the piecewise definition.

Try the absolute value function...

>> absoluteValue( -22 )
absX =
    22
ans =
    22
>> absoluteValue( 2 )
absX =
     2
ans =
     2

Fibonacci Numbers: Give a piecewise definition for the Fibonacci numbers.

Definition

• F(n) = 0 if n = 0
• F(n) = 1 if n = 1
• F(n) = F(n-2) + F(n-1) if n ≥ 2

Try coding it in Matlab.

function [ F ] = fibonacci( n )
    %FIBONACCI. Compute the n-th Fibonacci number.
    if n > 1
        F = fibonacci( n-1 ) + fibonacci( n-2 );
    elseif n == 1
        F = 1;
    else
        F = 0;
    end
end

Some examples of it running...

>> fibonacci( 0 )
ans =
     0
>> fibonacci( 1 )
ans =
     1
>> fibonacci( 2 )
ans =
     1
>> fibonacci( 3 )
ans =
     2
>> fibonacci( 4 )
ans =
     3
>> fibonacci( 5 )
ans =
     5

Notice that this function was also a straightforward translation of the piecewise definition into Matlab, even including the places where the mathematical definition referred to other values of the function being defined (something called “recursion” or a “recursive definition”).

“If” Lab

See the handout for details.

Next

Work on lab (I’ll also bring an extra credit extension to it).

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