How to find the domain of a square root function?
Understand the Problem
The question is asking how to determine the set of values (domain) for which a square root function is defined. This involves identifying the conditions under which the expression within the square root is non-negative.
Answer
$[3, \infty)$
Answer for screen readers
The domain is given by the inequality $f(x) \geq 0$ solved, expressed in interval notation.
Steps to Solve
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Identify the Expression Inside the Square Root
The domain of a square root function is determined by the expression inside the square root. Let's denote this expression as $f(x)$. For example, if the function is $\sqrt{x - 3}$, the expression inside the square root is $x - 3$.
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Set the Expression Inside the Square Root to be Non-Negative
For the square root function to be defined, the expression inside the square root must be greater than or equal to zero. Thus, set up the inequality $f(x) \geq 0$. For $\sqrt{x - 3}$, this becomes $x - 3 \geq 0$.
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Solve the Inequality
Solve the inequality to find the domain of the function. For $x - 3 \geq 0$, solve for $x$:
$$x \geq 3$$
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Write the Domain in Interval Notation
Express the solution to the inequality in interval notation. For $x \geq 3$, the domain is $[3, \infty)$.
Verify by substituting values to check if they satisfy the inequality.
The domain is given by the inequality $f(x) \geq 0$ solved, expressed in interval notation.
More Information
The concept of domains is crucial for understanding which inputs are valid for a given function. Square root functions, in particular, are only defined when the input is non-negative.
Tips
A common mistake is forgetting that the expression inside the square root must be greater than or equal to zero—not just greater—so you must include the boundary where the expression equals zero.
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