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How to find the domain of $g(x)=\sqrt[3]{x+3}$?

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Last updated date: 19th Jun 2024
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Answer
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Hint:For a real valued function, the domain of the function is a set of real numbers which consists of all the values of x for which the function yields a real value of y.For finding the domain of the given function take a cube on both sides of the function.

Complete step by step answer:
Let us first understand what is meant by domain of a function. Suppose, we have a function f such that $y=f(x)$, where x is the independent variable and y is the dependent variable that depends on the value of x. Then, we define something called the domain of the given function. For a real valued function, the domain of the function is a set of real numbers which consists of all the values of x for which the function yields a real value of y.
In other words, domain is the set of real values of x for which the value of y exists. In the given question, the function is g(x). And it is said that $g(x)=\sqrt[3]{x+3}$. Let us write that $g(x)=y$ for easy understanding.Hence, we get that $y=\sqrt[3]{x+3}$ …. (i)
Now, we have to find the values of x for which we have a real value of y. What we can do here is that we can take the cube of both sides of equation (i).
${{y}^{3}}={{\left( \sqrt[3]{x+3} \right)}^{3}}$
This further simplifies to ${{y}^{3}}=x+3$
Let us now analyse the left hand side of the above equation, i.e. ${{y}^{3}}$ .We know that a cube of any real number can yield a positive real number as well as a negative real number. It also yields a zero if the number is zero. Therefore, the value of ${{y}^{3}}$ can be a real positive number as well as a real negative number. It can also be equal to zero.
This means that the term ${{y}^{3}}$ takes the value of all the real numbers. Since ${{y}^{3}}=x+3$, then this means that $x+3$ takes the value of all the real numbers.We know that 3 is a real number and sum of two real numbers is a real number. Therefore, x can be any real number.

Hence, the domain of the given function is a set of all the real numbers.

Note:Note that no functions have domain of all the real numbers. For example, consider $y=\sqrt{x}$.Here, we can write that ${{y}^{2}}=x$. We know that for real values of y, ${{y}^{2}}$ is always positive or zero. Therefore, x must be zero or any positive real number. Hence, the domain of this function is a set of positive real numbers including zero.