6.1.3  Defining functions with boolean tests: ifte ?: when

You can use boolean tests to define functions not given by a single simple formula. Notably, you can use the ifte command or ?: operator to define piecewise-defined functions.

• ifte takes three arguments:
  • condition, a boolean condition.
  • true-result, the result to return if condition is true.
  • false-result, the result to return if condition is false.
• ifte(condition, true-result, false-result) returns true-result if condition is true and returns false-result if condition if false.

Example.
You can define your own absolute value function with:
Input:

Afterwards, entering:
Input:

will return:

However, myabs will return an error if it can’t evaluate the condition.
Input:

Output:

The ?: construct behaves similarly to ifte, but is structured differently and doesn’t return an error if the condition can’t be evaluated.

• The ?: construct takes three arguments:
  • condition, a boolean condition.
  • true-result, the result to return if condition is true.
  • false-result, the result to return if condition is false.
• condition?true-result:false-result returns true-result if condition is true and returns false-result if condition if false.

Example.
You can define your absolute value function with

If you enter

you will again get

but now if the conditional can’t be evaluated, you won’t get an error.
Input:

Output:

The when and IFTE commands are prefixed synonyms for the ?: construct.

• when (and IFTE) take three arguments:
  • condition, a boolean condition.
  • true-result, the result to return if condition is true.
  • false-result, the result to return if condition is false.
• when(condition, true-result, false-result) (and IFTE(condition, true-result, false-result)) return true-result if condition is true and returns false-result if condition if false.

and

all represent the same expression.

If you want to define a function with several pieces, it may be simpler to use the piecewise function.

• piecewise takes an unspecified (odd) number of arguments:
  • cond₁, return₁, cond₂, return₂, …, cond_n, return_n, an arbitrary number of pairs of conditions and corresponding return values.
  • default, a result to return if none of the conditions are true.
• piecewise(cond₁, return₁, …, cond_n, return_n, default) returns return_k if cond_k is the first true condition, or default if none of the conditions are true.

Example.
To define

you can enter:
Input: