Find gof and fog when f : R → R and g : R → R are defined by


Find gof and fog when $f: R \rightarrow R$                      and $g: R \rightarrow R$ are defined by

(i) $f(x)=2 x+3 \quad$                                                        and $\quad g(x)=x^{2}+5$

(ii) $f(x)=2 x+x^{2} \quad$                                                and $\quad g(x)=x^{3}$

(iii) $f(x)=x^{2}+8 \quad$                                                  and $\quad g(x)=3 x^{3}+1$

(iv) $f(x)=x \quad$                                                             and $\quad g(x)=|x|$

(v) $f(x)=x^{2}+2 x-3$                                                       and $g(x)=3 x-4$

(vi) $f(x)=8 x^{3} \quad$                                                    and $\quad g(x)=x^{1 / 3}$


Given, $f: R \rightarrow R$ and $g: R \rightarrow R$

So, gof: $R \rightarrow R$ and fog: $R \rightarrow R$

(i) $f(x)=2 x+3$ and $g(x)=x^{2}+5$

Now, (gof) $(X)$


$=g(2 x+3)$

$=(2 x+3)^{2}+5$

$=4 x^{2}+9+12 x+5$

$=4 x^{2}+12 x+14$

$(f \circ g)(x)$




$=2 x^{2}+10+3$

$=2 x^{2}+13$

(ii) $f(x)=2 x+x^{2}$ and $g(x)=x^{3}$

$(g \circ f)(x)$


$=g\left(2 x+x^{2}\right)$

$=\left(2 x+x^{2}\right)^{3}$

$(f o g)(x)$




$=2 x^{3}+x^{6}$

(iii) $f(x)=x^{2}+8$ and $g(x)=3 x^{3}+1$

$(g \circ f)(x)$




$(f o g)(x)$


$=f\left(3 x^{3}+1\right)$

$=\left(3 x^{3}+1\right)^{2}+8$

$=9 x^{6}+6 x^{3}+1+8$

$=9 x^{6}+6 x^{3}+9$

(iv) $f(x)=x$ and $g(x)=|x|$

$(g o f)(x)$




$(f o g)(x)$




(v) $f(x)=x^{2}+2 x-3$ and $g(x)=3 x-4$

(gof) (x)


$=g\left(x^{2}+2 x-3\right)$

$=3\left(x^{2}+2 x-3\right)-4$

$=3 x^{2}+6 x-9-4$

$=3 x^{2}+6 x-13$

$(f o g)(x)$


$=f(3 x-4)$

$=(3 x-4)^{2}+2(3 x-4)-3$

$=9 x^{2}+16-24 x+6 x-8-3$

$=9 x^{2}-18 x+5$

(vi) $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$

$(g \circ f)(x)$


$=g\left(8 x^{3}\right)$

$=\left(8 x^{3}\right)^{\frac{1}{3}}$

$=\left[(2 x)^{3}\right]^{\frac{1}{3}}$

$=2 x$

$(f \circ g)(x)$




$=8 x$








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