Question

Please help me with this problem​

Answer 1

`(f\circ g)(x)=(g\circ f)(x) \quad\text{only for x = 1}`

Explanation:

It can help to look at the difference of the two functions.

y1 = f(g(x)) = (x -1)^2

y2 = g(f(x)) = x^2 -1

The difference of the functions is ...

y1 -y2 = (x -1)^2 -(x^2 -1)

= x^2 -2x +1 -x^2 +1

= 2 -2x

This is a line with negative slope and an x-intercept at x=1. It means one function is neither always greater than nor always less than the other, and the two functions will be equal only at one point: x = 1.

`\bf{(f\circ g)(x)=(g\circ f)(x) \quad\textbf{only for x = 1}}`