Question

Let f : R → R and g : R → R be defined by f(x) = x² and g(x) = x + 1. Show that fog ≠ gof.

Answer 1

Given, f : R → R and g : R → R.
So, the domains of f and g are the same.

$$\begin{gathered} \left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)=f\left(x+1\right)={\left(x+1\right)}^2=x^2+1+2x \\ \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x^2\right)=x^2+1 \end{gathered}$$
So, fog ≠ gof