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How do you find the domain of the function $f\left( x \right)=\sqrt{x+8}$?

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Last updated date: 13th Jun 2024
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Answer
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Hint: We try to express the function and find the part which can make the expression invalid or undefined. We find the points on which the function inside the root part becomes equal to 0. Those points will be excluded from the domain of the expression $f\left( x \right)=\sqrt{x+8}$. The function inside the root has to be greater than or equal to 0.

Complete step-by-step solution:
We need to find the domain of the expression $f\left( x \right)=\sqrt{x+8}$.
The condition being that the expression has to give a defined solution.
The function inside the root part of the fraction $f\left( x \right)=\sqrt{x+8}$ has to be defined. The root part has to be greater than or equal to 0 otherwise complex roots will arrive.
The mathematical expression will be $x+8\ge 0$.
We subtract the value $-8$ from the inequation $x+8\ge 0$.
The inequation will be considered as an equation to perform the binary operations.
$\begin{align}
  & x+8\ge 0 \\
 & \Rightarrow x+8-8\ge 0-8 \\
 & \Rightarrow x\ge -8 \\
\end{align}$
We get the values of the variable $x$ which gives the solution range for the inequation $f\left( x \right)=\sqrt{x+8}$. The domain of the expression will be \[\left[ -8,\infty \right)\]. Therefore, \[\mathbb{R} - \left[ -8,\infty \right)\] will make the expression $f\left( x \right)=\sqrt{x+8}$ invalid

Note: We need to remember that \[\mathbb{R} - \left[ -8,\infty \right)\] is solely responsible for the expression to be undefined. We also considered that complex values as the root are not allowed. But if they are allowed then the domain becomes the whole real line.