  Question

If $$f(x)=\sqrt{x+3}$$ and $$g(x)={x}^{2}+1$$ be two real functions, then find $$f\circ g$$ and $$g\circ f$$.

Solution

$$f(x)=\sqrt{x+3}$$For domain, $$x+3\ge 0$$$$\Rightarrow$$  $$x\ge -3$$Domain of $$f=[-3,\infty)$$Range of $$f=(-3,\infty)$$Similarly, range of $$g=(1,\infty)$$Then, rangle of f is subset of domain $$g$$ and range of $$g$$ is subset of $$f$$.$$\therefore$$  $$f\circ g$$ and $$g\circ f$$ exist.$$\Rightarrow$$  $$f\circ g(x)=f[g(x)]$$                     $$=f(x^2+1)$$  ......... [ Since, $$g(x)^2+1$$ ]                     $$=\sqrt{x^2+1+3}$$ ......... [ Since, $$f(x)=\sqrt{x+3}$$ ]                     $$=\sqrt{x^2+4}$$$$\Rightarrow$$  $$g\circ f(x)=g[f(x)]$$                      $$=g(\sqrt{x+3})$$.........  [ Since, $$f(x)=\sqrt{x+3}$$ ]                      $$=(\sqrt{x+3})^2+1$$......... [ Since, $$g(x)^2+1$$ ]                      $$=x+3+1$$                      $$=x+4$$Maths

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