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.
1 | \substitute[<action>]{<variable>}{<expression>}{<free_variable>}{<substitute>}
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- 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
- 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
1 2 3 | \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
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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}