given h(x)= 4+1/ √ x-5 find the functions f and g that h(x)= (f o g)(x)

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ishpiro's profile pic

ishpiro | College Teacher | (Level 1) Educator

Posted on

Note that in this question, as in most questions involving the decomposition of a function, f and g are not unique. Other possible choices for f and g might be

`f = 4 + x` and `g = 1/sqrt(x-5)`

or

`f = 4 + 1/x` and `g = sqrt(x-5)`

Both of these pairs of f and g will result in the composition function

`f o g (x) = 4+1/sqrt(x-5)`

e-devam's profile pic

e-devam | High School Teacher | (Level 2) Adjunct Educator

Posted on

Here, we have to do the decomposition of the given function,

`h(x)= 4+1/sqrt(x-5)` into `f(x)` and `g(x)` such that `h(x)=(fog)(x)` .

In `(f o g)(x)` , `g(x)` is the inner function and `f(x)` is the outer function.

`g(x)` is to be plugged into `f(x)` to get `(fog)(x)` .

Hence, `g(x)= (x-5)` and

`f(x)=4+1/sqrtx`

Sources:
mathisthebest's profile pic

mathisthebest | eNotes Newbie

Posted on

Hi, just wondering but can't f=4+1/(x-2)^0.5 and g=x-3 as well?

Doesn't this solution exist as well?

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mathisthebest's profile pic

mathisthebest | eNotes Newbie

Posted on

So wouldn't there be an infinite number of solutions?

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