Question

Consider the functionf:R→Rdefined viaf(x) =|x|.(a) Give a functiongwith domainRsuch thatg◦fis one-to-one, or describe why it is not possible.(b) Give a function with domain such that◦fis onto, or describe why it is not possible.(c) Give a functiongwith rangeRsuch thatf◦gis one-to-one, or describe why it is not possible.(d) Give a functiongwith rangeRsuch thatf◦gis onto, or describe why it is not possible.

Answer 1

(a) Is not possible

(b) It is possible

(c) It is possible

(d) Is NOT possible.

Explanation:

(a)

Is not possible, notice that for any function
`g` such that

`g : \mathbb{R} \rightarrow \mathbb{R}`

you would have that

`(g\circ f)(x) = g(f(x)) = g(|x|)`

And for, lets say -3,3 you have that

`g(|-3|) = g(|3|) = g(3)` therefore is not possible to find a function that is one to one.

(b)

It is possible. Take the following function

`g(x) = x\sin(x)` since
`\sin` is periodic it will take positive and negative numbers and if you multiply by
`x` each period will become bigger and bigger.

(c)

It is possible. Take the function

`g(x) = √(x)`

Then

`(f \circ g )(x) = | √(x) | = √(x)` and
`√(x)` is one to one.

(d)

It is NOT possible because
`(f\circ g)(x) = f(g(x)) = |g(x)|` and that will always be positive.