Respuesta :
SOLUTION
The function given is
[tex]f(x)=\sqrt{7-x^2}[/tex]The domain of a function is the set of input values (x-values) for which the function is defined or real.
To obtain the domain of the function above, we need to solve the expression in the square root.
[tex]\begin{gathered} 7-x^2\ge0 \\ \text{Then, subtract 7 from both sides} \\ 7-7-x^2\ge0-7 \\ -x^2\ge-7 \\ \text{Multiply both sides by -1} \\ x^2\le7 \end{gathered}[/tex]Take square root of both sides we have
[tex]\begin{gathered} \sqrt{x^2}\le\pm\sqrt[]{7} \\ \text{Then} \\ x^{}\le\pm\sqrt[]{7} \end{gathered}[/tex]Hence, the domain becomes
[tex]\begin{gathered} -\sqrt{7}\le\: x\le\sqrt{7} \\ or \\ \mleft[-\sqrt{7},\: \sqrt{7}\mright] \end{gathered}[/tex]Domain is [-√7,√7]
Similarly, for th range of f(x) we have
[tex]\begin{gathered} \mleft[0,\: \sqrt{7}\mright] \\ or \\ \: 0\le\: f\mleft(x\mright)\le\sqrt{7} \end{gathered}[/tex]Therefore
Range is [0,√7]
