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 of) (ii) (f o g)

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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 of) (ii) (f o g)

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 of) (ii) (f o g)

Solution:

(i) g of
We need to find: $g \circ f$ .
Formula used: $g$ o $f=g(f(x))$
It is given that: (i) $f: R \rightarrow R: f(x)=(2 x+1)$
(ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$
Solution: We have,
$\begin{array}{l} g \circ f=g(f(x))=g(2 x+1)=\left[(2 x+1)^{2}-2\right] \\ \Rightarrow 4 x^{2}+4 x+1-2 \\ \Rightarrow 4 x^{2}+4 x-1 \end{array}$
Ans). $g$ of $(x)=4 x^{2}+4 x-1$

(ii) $f \circ g$
We need to find: $\mathrm{f} \circ \mathrm{g}$
Formula used: f o $g=f(g(x))$
It is given that: (i) $f: R \rightarrow R: f(x)=(2 x+1)$
(ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$
Solution: We have,
$\begin{array}{l} \text { fo } g=f(g(x))=f\left(x^{2}-2\right)=\left[2\left(x^{2}-2\right)+1\right] \\ \Rightarrow 2 x^{2}-4+1 \\ \Rightarrow 2 x^{2}-3 \end{array}$
Ans). fo $g(x)=2 x^{2}-3$