Please help me to answer and understand this question

Question

Question

-- What happens if we graph both f and f ^{– 1} on the same set of axes, using the x-axis for the input to both f and f ^{– 1}? 

[Suggestion:

go to desmos/calculator and type y = x³ {-2 < x < 2}, y = x ^1/3 {-2 < x < 2} and y = x {-2 < x < 2}, and

1. describe the relationship between the three curves.]
2. Then write your own example discussing the difficulty of graph both f and f ^{– 1} on the same set of axes. 

--Suppose f = R – R is a function from the set of real numbers to the same set with f(x) = x + 1.

We write f² to represent f o f and f ^{n + 1} = f ⁿ o  f.

1. Is it true that f² o f = f o f² ? Why?
2. Is the set {g: R – R l g o f = f o g }infinite? Why? 

Your Discussion should be a minimum of 250 words.

Answer 1

(1) Function and it's inverse are symmetric about the line y=x

(2)

f(f(x)) = (x+1)+1 = x+2

f_3(x) = (x+1)+2 = x+3

f_4(x) = (x+1)+3 = x+4

...

by induction... suppose f_n(x) = x+n

f_(n+1) = (x+1)+n = x+(n+1)

vs

f_2 comp f = (x+1)+2 = x+3

f_3 comp f = (x+3)+1 = x+4

....

again by induction...

if f_n = x+n then f_n comp f = (x+1)+n = x+(n+1)

therefore f_n comp f = f comp f_n = f_(n+1) = x +(n+1)

because

f comp f_n =

(x+n)+1 = x+(n+1) <--- associative

= x+(1+n) <--- commutative

= (x+1)+n <--- associative

= f_n comp f