How to find these function compositionWe have 3 functions, f,g,h:R->R, where f(x)=|x|, g(x)=6x-8, h(x)=4-x u=?, where u=f*[(g*f)/h], "*" means composition of function

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to compose the functions `f(x)` and `g(x)` using the relation below, such that:

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

Reasoning by analogy, you need to substitute the equation of f(x) for x, in equation of function g(x), such that:

`g(f(x))= g(|x|) =>g(f(x)) = 6|x| - 8 `

You need to divide `g(f(x))` by `h(x) = 4 - x` , such that:

`(g(f(x)))/(h(x)) = (6|x| - 8)/(4 - x)`

You need to come up with the following substitution, such that:

`(g(f(x)))/(h(x)) = v(x)`

You need to evaluate the final composition of functions, such that:

`(fov)(x) = f(v(x)) => (fov)(x) = |v(x)|`

You need to substitute the equation of `v(x)` such that:

`(fov)(x) = |(6|x| - 8)/(4 - x)| => {((fov)(x) = (6x-8)/(4-x), for (6x-8)/(4-x)>0, x>=0),((fov)(x) = (8-6x)/(4-x), (6x-8)/(4-x)>0, x<0),((fov)(x) = -(6x-8)/(4-x), (6x-8)/(4-x)<0, x>=0),((fov)(x) = -(8-6x)/(4-x), (6x-8)/(4-x)<0, x<0):}`

Hence, evaluating the composition of the given functions, yields `{((fov)(x) = (6x-8)/(4-x), for (6x-8)/(4-x)>0, x>=0),((fov)(x) = (8-6x)/(4-x), (6x-8)/(4-x)>0, x<0),((fov)(x) = -(6x-8)/(4-x), (6x-8)/(4-x)<0, x>=0),((fov)(x) = -(8-6x)/(4-x), (6x-8)/(4-x)<0, x<0):}.`

giorgiana1976 | Student


(g*f)/h=(6*|x|-8)/(4-x), for x different from 4


u=|(6*|x|-8)/(4-x)| ,where u:R\{4}->R