Defining functions with boolean tests

### 5.1.3 Defining functions with boolean tests

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

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

#### Example

You can define your own absolute value function with:

myabs(x):=ifte(x>=0,x,-x)

Hence:

myabs(-4)

4 |  |  |  |  |  |  |  |  |  |

However, myabs will return an error if it cannot evaluate the condition.

myabs(x)

Ifte: Unable to check test Error: Bad Argument Value |  |  |  |  |  |  |  |  |  |

The ?: construct behaves similarly to ifte, but is structured differently and does not return an error if the condition cannot be evaluated.

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

#### Example

You can define your absolute value function with

myabs(x):=(x>=0)?x:-x

If you enter

myabs(-4)

you will again get

4 |  |  |  |  |  |  |  |  |  |

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

myabs(x)

x≥ 0 ? x : −x |  |  |  |  |  |  |  |  |  |

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

- when (and IFTE) take three arguments:
  - condition, a boolean condition.
  - restrue, the result to return if condition is true.
  - resfalse, the result to return if condition is false.
- when(condition,restrue,resfalse) and IFTE(condition,restrue,resfalse) both return restrue if condition holds and resfalse otherwise.

(condition) ? restrue : resfalse
when(condition,restrue,resfalse)

and

IFTE(condition,restrue,resfalse)

all represent the same expression.

If you want to define a function with several pieces, it may be simpler to use the piecewise function. The latter can also be used for finding piecewise representations of expressions.

- piecewise can take an even number of arguments, followed by a single optional argument:
  - cond1, return1, cond2, return2, …, condn, returnn, an arbitrary number of pairs of conditions and corresponding return values.
  - Optionally, expr, an expression.
  Alternatively, piecewise can take one argument, an expression expr, optionally followed by a second argument x, an identifier.
- With conditional arguments, piecewise(cond1,return1,…,condn,returnn ⟨,expr ⟩) returns an expression which evaluates to returnk if condk is the first true condition, or to expr if none of the conditions are true. If expr is not given, then the default value is undef.
- With a single argument, piecewise(expr) returns a piecewise representation of expr. This is useful if expr involves Heaviside/signum/absolute value functions, since piecewise attempts to remove them; if it fails, then expr is returned. The optional second argument may be given, specifying the relevant variable (otherwise it is guessed).

#### Examples

To define

f(x)= |
⎧ ⎪ ⎨ ⎪ ⎩ |
−2 | if x < −2
3x+4 | if −2 ≤ x < −1
1 | if −1 ≤ x < 0
x+1 | if x ≥ 0

enter:

f(x):=piecewise(x<-2,-2,x<-1,3*x+4,x<0,1,x+1)

To verify, plot f(x) for e.g. −3≤ x≤ 1.

To convert the expression x θ(x+1)−x2 θ(x−1)+x3 sign(x) to a piecewise expression, enter:

f:=x*Heaviside(x+1)-x^2*Heaviside(x-1)+x^3*sign(x):;

and then

piecewise(f)

or:

piecewise(f,x)

⎧ ⎪ ⎨ ⎪ ⎩ |
−x3, | −1>x
−x3+x, | 0>x
x3+x, | 1>x
x3−x2+x, | otherwise

piecewise can be used for “flattening” expressions containing piecewise defined subexpressions. For example:

piecewise(1+piecewise(x<0,-x,x<1,x,1)*when(x<1/2,4x^2,2-2x),x)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ |
−4 x3+1, | 0>x
4 x3+1, |
1
2
| >x
−2 x2+2 x+1, | 1>x
−2 x+3, | otherwise