Which of the function in exercise 3 of Section 51 are onetoo

Which of the function in exercise 3 of Section 5.1 are one-to-one? Which are onto? Which of the function in exercise 4 of Section 5.1 are one-to-one? Which are onto? Let A = R \\ {1}, and let f: A rightarrow A be defined as follows: f(x) = x + 1/x - 1 Show that f is one-to-one and onto. Show that f compositefunction f = i_A. Let A = P(R). Define f; R rightarrow A by the formula f(x) = {y elementof R} y^2

Solution

a.

Let, c be in C ,gof is onto so there is some a in A so that

gof(a)=c

g(f(a))=c

f(a) is in B

SO there is some b=f(a) in B so that

g(b)=c

Hence, g is onto

b)

Let, f(a)=f(b)

HEng, g(f(a))=g(f(b))

gof is one to one

So, a=b

Hence, f is one to one