Substituting Functions: Simplifying Multivariate Expressions

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In summary, substituting functions is the process of replacing one function with another function that has the same output for a given input. The steps for substituting functions include identifying input and output variables, finding corresponding variables in the replacement function, replacing the input variable, and simplifying the expression. Not all functions can be substituted with each other, as they must have the same domain and range. Substituting functions is useful for simplifying complex expressions, solving equations, finding inverse functions, and transforming functions. The order in which functions are substituted does not affect the final result, but it is important to carefully replace all instances of the input variable in the original function.
  • #1
fred2028
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This is more of a concept question, so the template is not followed.

Say you're given

T(x,y,z) = xy-z

And

z = x+y

Basically, T is a function of x, y, and z while z is a function of x and y. If we substitute z into T, would T become T(x, y) or stay T(x, y, z)?
 
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  • #2
Your function would become T(x,y)
 
  • #3
rock.freak667 said:
Your function would become T(x,y)

OK thanks!
 

What is "Substituting functions"?

Substituting functions is a mathematical process where one function is replaced by another function with the same output for a given input. This is useful in simplifying complex expressions and solving equations.

What are the steps for substituting functions?

The general steps for substituting functions are:

  1. Identify the input and output variables in the function being substituted.
  2. Find the corresponding input and output variables in the replacement function.
  3. Replace the input variable in the original function with the replacement function.
  4. Simplify the resulting expression.

Can any function be substituted with another function?

No, in order for a function to be substituted with another function, they must have the same output for a given input. This means that the functions must have the same domain and range.

When is substituting functions useful?

Substituting functions is useful in simplifying complex expressions, solving equations, and finding the inverse of a function. It can also be used to transform functions, such as shifting or stretching them.

Is there a specific order in which functions should be substituted?

The order in which functions are substituted does not affect the final result. However, it is important to carefully identify and replace all instances of the input variable in the original function with the replacement function.

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