Substitute¶

Creates a new function by replacing a free variable by an earlier defined variable. Main purposes are to obtain the composition of two functions or to evaluate a function at a given value.

  \substitute[<action>]{<variable>}{<expression>}{<free_variable>}{<substitute>}

action, variable and expression are identical to the input of a \function command.
free_variable defines the free variable in the given expression.
substitute must be an earlier defined variable. The free_variable will be replaced with the substitute variable.
\(h = f \circ g\) is the output of

  \substitute{h}{f}{x}{g}

This command can also be used to determine the value of a given function at a specific point \(x_0\) (using a constant function variable). For example, \(6^4 - 5 \cdot 6^3 + 4 \cdot 6^2 + 3 \cdot 6 + 7\) could be the output of

  \function{f}{x^4 - 5x^3 + 4x^2 + 3x + 7}    % defines a function named f
  \number{x0}{6}                          % point where f is to be evaluated
  \substitute{fx0}{f}{x}{x0}                  % this is the function f at point x_0

Be aware that you cannot directly substitute x by a literal number e.g. 2, but that you need a variable taking this value, e.g. by using \number{two}{2} and \substitute{f2}{f}{x}{two}.

You can also normalize, expand, sort and calculate the \derivative and \substitute command. They take the same option action as a normal \function. In fact, you would often want to normalize a derivative, compare the following output with the earlier example shown for derivative. \(f_1 = 4x^3 + 6x^2\) is the output of

\derivative[normalize]{f_1}{x^4 + 2*x^3}{x}