Question

Lea says that (f°g)(x)=(g°f)(x). Do you agree with her or not? Explain.

Answer 1

No, because you substitute differently in both so they are not equal.

Answer 2

Answer with explanation:

No, Lea is incorrect.

" The composition of two functions may be different if they are taken in different orders "

For example:

If,

`f(x)=x+2`

and

`g(x)=x^2`

Then,

`(fog)(x)=f(g(x))\\\\(fog)(x)=f(x^2)\\\\(fog)(x)=x^2+2`

Also,

`(gof)(x)=g(f(x))\\\\(gof)(x)=g(x+2)\\\\(gof)(x)=(x+2)^2\\\\(gof)(x)=x^2+4x+4`

Now, we know that (gof)(x) and (fog)(x) will not be equal for all the values of x.

They will be equal only for x= -1/2

( Since,

`x^2+2=x^2+4x+4\\\\2=4x+4\\\\4x=2-4\\\\4x=-2\\\\x=(-2)/(4)\\\\x=(-1)/(2)`