elementary functions and non-elementary functions

[shahar]

1. Is a graph of elementary function can be build by non-elementary function. i.e. the graphs of the functions are the same?
2. Is the opposite is available?

 

[pka]

1. Is a graph of elementary function can be build by non-elementary function. i.e. the graphs of the functions are the same?
2. Is the opposite is available?

Pleas define your terms: elementary function and non-elementary function.

 

[shahar]

elementary function is: cos(x), sin(x), etc. i.e. simple functions.
non-elementary function isn't simple functions.

 

[Dr.Peterson]

1. Is a graph of elementary function can be build by non-elementary function. i.e. the graphs of the functions are the same?
2. Is the opposite is available?

Let's take this definition:

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots). [See also here.]​

Now the question is, what do you mean by "build" (i.e. built)? Do you mean, is it possible to form a function in that list by composition of functions none of which (or only some of which) are in that list? Then certainly the answer is yes; the composition of any function and its inverse is an elementary function!

And why do you mention graphs at all? Two functions whose graphs are identical are the same function.

When we get to your second question, you'll have to explicitly state what you mean by "the opposite".

 

[shahar]

question #2: Is a graph of non-elementary function can be build by elementary function. i.e. the graphs of the functions are the same?

 

[Dr.Peterson]

Look at the definition I provided! Any function that can be "built" (by composition) from elementary functions is, by definition, an elementary function. It can't be "non-elementary".