Mathematics - Functions Question with Solution | TestHub

Given $\mathrm{f}(\mathrm{x}+\mathrm{y}+1)=(\sqrt{\mathrm{f}(\mathrm{x})}+\sqrt{\mathrm{f}(\mathrm{y})}{)}^2$. Putting $\mathrm{y}=0$, we get $\mathrm{f}(\mathrm{x}+1)=(\sqrt{\mathrm{f}(\mathrm{x})}+1{)}^2$.

Putting $x=0$, we have $\mathrm{f}(1)=(\sqrt{\mathrm{f}(0)}+1{)}^2=(1+1{)}^2=2^2=4$.

Putting $x=1$, $\mathrm{f}(2)=(\sqrt{\mathrm{f}(1)}+1{)}^2=(\sqrt{4}+1{)}^2=(2+1{)}^2=3^2=9$.

Putting $x=2$, $\mathrm{f}(3)=(\sqrt{\mathrm{f}(2)}+1{)}^2=(\sqrt{9}+1{)}^2=(3+1{)}^2=4^2=16$.

Proceeding in this way, we get $\mathrm{f}(\mathrm{x})=(\mathrm{x}+1{)}^2$.