# Solve this

Question:

Let $f: R \rightarrow R: f(x)=(2 x+1)$ and $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$

Write down the formulae for

(i) (g o f)

(ii) (f o g)

(iii) (f o f)

(iv) (g o g)

Solution:

(i) g o f

To find: g o f

Formula used: g o f = g(f(x))

Given: (i) f : R → R : f(x) = (2x + 1)

(ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$

Solution: We have,

$g \circ f=g(f(x))=g(2 x+1)=\left[(2 x+1)^{2}-2\right]$

$\Rightarrow 4 x^{2}+4 x+1-2$

$\Rightarrow 4 x^{2}+4 x-1$

Ans). $g \circ f(x)=4 x^{2}+4 x-1$

(ii) f o g

To find: f o g

Formula used: f o g = f(g(x))

Given: (i) f : R → R : f(x) = (2x + 1)

(ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$

Solution: We have,

$f \circ g=f(g(x))=f\left(x^{2}-2\right)=\left[2\left(x^{2}-2\right)+1\right]$

$\Rightarrow 2 x^{2}-4+1$

$\Rightarrow 2 x^{2}-3$

Ans). $f \circ g(x)=2 x^{2}-3$

(iii) $f$ o $f$

To find: f o f

Formula used: f o f = f(f(x))

Given: (i) f : R → R : f(x) = (2x + 1)

Solution: We have,

$f \circ f=f(f(x))=f(2 x+1)=[2(2 x+1)+1]$

$\Rightarrow 4 x+2+1$

$\Rightarrow 4 x+3$

Ans). $f$ of $f(x)=4 x+3$

(iv) $\mathrm{g} \circ \mathrm{g}$

To find: g o g

Formula used: g o g = g(g(x))

Given: (i) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$

Solution: We have,

$g \circ g=g(g(x))=g\left(x^{2}-2\right)=\left[\left(x^{2}-2\right)^{2}-2\right]$

$\Rightarrow x^{4}-4 x^{2}+4-2$

$\Rightarrow x^{4}-4 x^{2}+2$

Ans). $g \circ g(x)=x^{4}-4 x^{2}+2$