# If f(x) = 2x + 5 and

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) $f^{2}(x)=f(x) \times f(x)$

$=(2 x+5)(2 x+5)$

$=(2 x+5)^{2}$

$=4 x^{2}+20 x+25$