Question:
If $f(x)=2 x+5$ and $g(x)=x^{2}+1$ be two real functions, then describe each of the following functions:
(i) fog
(ii) gof
(iii) fof
(iv) f2
Also, show that $f \circ f \neq f^{2}$
Solution:
f(x) and g(x) are polynomials.
$\Rightarrow f: R \rightarrow R$ and $g: R \rightarrow R$.
So, $f \circ g: R \rightarrow R$ and $g \circ f: R \rightarrow R$.
(i) $(f \circ g)(x)=f(g(x))$
$=f\left(x^{2}+1\right)$
$=2\left(x^{2}+1\right)+5$
$=2 x^{2}+2+5$
$=2 x^{2}+7$
(ii) $(g \circ f)(x)=g(f(x))$
$=g(2 x+5)$
$=(2 x+5)^{2}+1$
$=4 x^{2}+20 x+26$
(iii) $(f o f)(x)=f(f(x))$
$=f(2 x+5)$
$=2(2 x+5)+5$
$=4 x+10+5$
$=4 x+15$
(iv) $f^{2}(x)=f(x) \times f(x)$
$=(2 x+5)(2 x+5)$
$=(2 x+5)^{2}$
$=4 x^{2}+20 x+25$