# give an example of a composite function. Be sure to explain (1) what your variables are, (2) how they are represented in the function,give an example of a composite function. Be sure to explain...

give an example of a composite function. Be sure to explain (1) what your variables are, (2) how they are represented in the function,

give an example of a composite function. Be sure to explain (1) what your variables are, (2) how they are represented in the function, and (3) which elementary functions are combined to form the composite function

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### 1 Answer

You need to remember how to compose two functions, such that:

`(fog)(x) = f(g(x))`

Notice that the variable x of the function `f(x) ` is replaced by the function `g(x), ` hence, `g(x)` becomes the variable of function `f(g(x)).`

Considering the function `f(x) = x - 1` and the function `g(x) = 2^x` , you may compose the functions `f(x)` and `g(x)` such that:

`(fog)(x) = f(g(x))`

You need to substitute `g(x)` for x in equation of the function `f(x)` such that:

`f(g(x)) = g(x) - 1`

Substituting `2^x` for `g(x)` yields:

`f(g(x)) = 2^x- 1`

`(gof)(x) = g(f(x))`

You need to substitute `f(x)` for x in equation of the function `g(x)` such that:

`g(f(x)) = 2^(f(x))`

Substituting `x - 1` for `f(x) ` yields:

`g(f(x)) = 2^(x-1) => g(f(x)) = (2^x)/2`

Notice that `(fog)(x) != (gof)(x).`

Composing `(fog)(x)` or `(gof)(x)` yields that the variables are either the function `g(x), ` or the function `f(x),` and the elementary functions involved are the linear function `f(x) = x - 1` and the exponential function `g(x) = 2^x` .

**Hence, evaluating the compositions of the functions considered yields `f(g(x)) = 2^x - 1` and `g(f(x)) = (2^x)/2` and `(fog)(x) != (gof)(x).` **