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- Expressions usable in \function
- Expressions usable in \checkFuncForZero
- Relations usable in \randadjustIf

In a MUMIE TeX problem document you may use the `\function`

command within a variables section to define a variable of type function.

*Syntax:*

`\function{ _function_name_ }{ _expression_ }`

*Example:*

1234 `\begin{variables}`

` `

`\number{a}{-2}`

` `

`\function{f}{a*x^2+2}`

`\end{variables}`

This introduces first a number variable $$a$$ with value $$-2$$, and then a function variable $$f$$ with value $$-2x²+2$$, considering that a is assigned the value $$-2$$.

More generally: An *expression* in a MUMIE problem is by definition:

*Definition*

- a fixed number from a field defined by
`\field`

- a variable name including variables (or functions) defined elsewhere in the
*variables*section. - if
*expr*is an expression, then so is (*expr*) - if
*expr*is an expression, then so is -*expr* - if
*expr1*and*expr2*are expressions, then $$*expr1*+*expr2*$$ is an expression - same for $$-$$ ,$$*$$ ,$$/$$ ,$$^$$, $$mod$$
- if
*expr*is an expression, then so is`sqrt( _expr_ )`

, denoting the square root- other admissible function names are:
`cbrt()`

-cubic root- $$sin()$$, $$cos()$$, $$tan()$$, $$cot()$$, $$sinh()$$, $$cosh()$$, $$tanh$$, $$coth()$$, $$arcsin()$$, $$arccos()$$, $$arctan()$$, $$arccot()$$,

$$arsinh()$$, $$arcosh()$$, $$artanh()$$, $$arcoth()$$, $$ln()$$, $$log_base()$$ base must be integer or Euler number $$e$$, $$log()$$, $$exp()$$ ,

with usual meaning - also $$abs()$$ for absolute value, $$fac()$$ for integer factorial function, $$floor()$$ math floor, $$re(), im()$$ and $$conj()$$ for complex real and imaginary part and complex conjugate,
- finally, $$sign()$$ and $$theta()$$ to denote the signum and Heaviside functions.

- if
*expr*is an expression, then so is`| _expr_ |`

as an alternative for`abs( _expr_ )`

- if
*expr*is an expression and n is a positive integer, than`expr_#n`

is an alternative way to write`expr^(1/n)`

*Hints:*

- observe the usual precedence rules: so
`_expr1_ * _expr2_`

is in general*not*the product of both expressions as seen by the example $$a+b*c+d$$ - $$mod$$ denotes the modulo operator; thus, for two integral numbers $$a$$ and $$b$$,
`a mod b`

is the remainder of the division of a by b - standard functions have to be written with their parentheses, so sin x is not permitted
- do not denote variables by standard function names
- observe that in the generic TeX framework discussed here all functions defined by the
`\function`

command are expected to define functions on $$R, N, Q$$ or, possibly, $$C$$,

so the expression in`function{ _function_name_ }{ _expression_ }`

should use at most one more variable than defined in the

corresponding*variables*section, namely the independent variable of the function @-placeholder-@, denoting an

element of the*function domain*. You may call*it*x or y or whatever, but this variable name must not occur in the*variables*section.

The `\checkFuncForZero`

command is used to check a functional of user defined functions for being zero numerically.

User defined functions are e.g. defined by the student's inputs via the command `\inputAsFunction`

. The corresponding inputs can be

checked by the corrector program using the criterion delivered by `\checkFuncForZero`

.

*Syntax:*

`\checkFuncForZero{ _functionial_expression_ }{ _lower_check_bound_ }{ _upper_check_bound_ }{ _number_checkpoints_ }`

A functional_expression is in principle an expression in the occuring functions which may be subject to the usual arithmetic

operations and additionally composition and derivation operations.

So the definition of an expression as above is extended by the following:

- any expression as defined above is a functional expression, too
- a function name declared by an
`\inputAsFunction`

command in the same question block is a functional expression - If
*expr*is a functional expression and $$f$$ is a function name either declared by a`\function`

command or by an`\inputAsFunction`

command in the same question block, then*f[*is a functional expression*expression*]

you must use *square brackets* in case of these function names, while you have to use *round brackets* in case of standard functions like $$sqrt$$ or $$sin$$

- If
*expr*is a functional expression, then*D[*is a functional expression, denoting the derivative of*expression*]

expression (so avoid $$D$$ as a functions or variables name)

This defines, what a syntactically correct functional expression is.

Hence in particular you may use

- fixed numbers as 3.1415
- valid number variables, declared by
`\number`

or`\randint`

- valid programmer defined functions declared by
`\function`

- valid user defined (input) functions declared by
`\inputAsFunction`

- explicitly defined functions (as x^2+3), but
*only x is allowed here*as valid (non-declared) 'true' variable (@-placeholder-@)

*Examples:*

`f[g]`

, if $$f$$ and $$g$$ are function names`sin(f)`

, if f is a function name`f[sin(x)]`

, if f is a function name`D[D[f]]`

, if f is a function name`f[x^3+a*x+b]`

, if f is a function's name and a is a declared number variable or function variable

The `\randadjustIf`

command is used to redefine number variables which had been declared by`\randint`

in order to avoid certain unfavorable combinations of variables.

*Syntax:*

`\randadjustIf{ _list_of_variables_ }{ _avoidance_relation_ }`

We define, what a relation is:

- If
*expr1, expr2*are expressions, then*expr1*=*expr2*is a relation - same for the other comparision operators $$!=$$ (means unequal), $$>$$, $$<$$, $$>=$$ , $$<=$$
- If
*rel*is a relation, then [*rel*] is a relation (observe that*square brackets*are used to bundle relations) - If
*rel1*,*rel2*are relations, then*rel1*AND*rel2*is a relation, as well as*rel1*OR*rel2*and

NOT*rel1*are relations

Observe the precedence rules: NOT precedes AND precedes OR.

Also, bear in mind, that we are talking about avoidance relations in the context of `\randadjustIf`

, so in the example

123 `\randint{a}{2}{20}`

`\randint[Z]{b}{-20}{20}`

`\randadjustIf{a,b}{a=b}`

the result are random parameters a and b which are different. Remember that the [Z]-flag instructs the compiler to

avoid zero for *b*. So this example yields the same result as

123 `\randint{a}{2}{20}`

`\randint{b}{-20}{20}`

`\randadjustIf{a,b}{a=b OR b=0}`

Updated by **Greiser**, **9 months ago **– 46cf241