No, Lea is incorrect.
" The composition of two functions may be different if they are taken in different orders "
For example:
If,
[tex]f(x)=x+2[/tex]
and
[tex]g(x)=x^2[/tex]
Then,
[tex](fog)(x)=f(g(x))\\\\(fog)(x)=f(x^2)\\\\(fog)(x)=x^2+2[/tex]
Also,
[tex](gof)(x)=g(f(x))\\\\(gof)(x)=g(x+2)\\\\(gof)(x)=(x+2)^2\\\\(gof)(x)=x^2+4x+4[/tex]
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,
[tex]x^2+2=x^2+4x+4\\\\2=4x+4\\\\4x=2-4\\\\4x=-2\\\\x=\dfrac{-2}{4}\\\\x=\dfrac{-1}{2}[/tex]
