Problem 42
Question
Find the domain of the expression. \(\sqrt{147-3 x^{2}}\)
Step-by-Step Solution
Verified Answer
The domain of the given expression is all real numbers x such that \( -7 \leq x \leq 7 \).
1Step 1: Set the Radicand Greater Than or Equal to Zero
To find the domain of the expression, set the radicand (the value under the square root) greater than or equal to zero: \(147 - 3x^2 \geq 0\). This is because the square root of a number is only real when the number is either positive or zero.
2Step 2: Rearrange the Equation to Isolate x
Rearrange the inequality to isolate x and solve for x: \(\frac{147}{3} \geq x^2 \). This simplifies to \( x^2 \leq 49 \), which means that x can be any number such that the square of the number is less than or equal to 49.
3Step 3: Solve for x
Since \( x^2 \leq 49 \) we can find the possible values of x by taking the square root of both sides. This gives us \( x \leq 7\) and \( x \geq-7 \).
Key Concepts
Radicand InequalitySolving Quadratic InequalitiesSquare Root Properties
Radicand Inequality
When it comes to understanding the domain of an expression, particularly those involving square roots, we need to pay close attention to the radicand inequality. The radicand is the term inside the square root symbol, and for a square root to be defined in real numbers, the radicand must be non-negative, that is greater than or equal to zero.
This rule arises from the fact that the square root of a negative number isn't a real number but rather an imaginary one. To determine the domain of the expression \(\sqrt{147-3 x^{2}}\), we set the radicand \(147 - 3x^2 \geq 0\), ensuring that the values we find for \(x\) will keep the expression within the real numbers.
It's imperative to understand that when we solve the inequality \(147 - 3x^2 \geq 0\), we're essentially determining the set of all possible \(x\) values that make our original expression a real number.
This rule arises from the fact that the square root of a negative number isn't a real number but rather an imaginary one. To determine the domain of the expression \(\sqrt{147-3 x^{2}}\), we set the radicand \(147 - 3x^2 \geq 0\), ensuring that the values we find for \(x\) will keep the expression within the real numbers.
It's imperative to understand that when we solve the inequality \(147 - 3x^2 \geq 0\), we're essentially determining the set of all possible \(x\) values that make our original expression a real number.
Solving Quadratic Inequalities
Dealing with quadratic inequalities can be tricky, but it's a crucial skill when defining the domain of expressions like \(\sqrt{147-3 x^{2}}\). The inequality \( x^2 \leq 49 \) is a simple form of a quadratic inequality, where we are looking for all the numbers whose squares are less than or equal to 49.
To solve this, we interpret it as \( x \leq 7\) and \( x \geq -7\), since 7 and -7 are the square roots of 49. The compound inequality signifies that our solution includes all the real numbers between -7 and 7, inclusive of these endpoints, which corresponds to the domain of our original expression.
It's essential to visualize this on a number line: imagine all points filled in between -7 and 7, including these two numbers. Any number outside this range would yield a negative radicand, which isn't allowed in the domain of a square root function.
To solve this, we interpret it as \( x \leq 7\) and \( x \geq -7\), since 7 and -7 are the square roots of 49. The compound inequality signifies that our solution includes all the real numbers between -7 and 7, inclusive of these endpoints, which corresponds to the domain of our original expression.
It's essential to visualize this on a number line: imagine all points filled in between -7 and 7, including these two numbers. Any number outside this range would yield a negative radicand, which isn't allowed in the domain of a square root function.
Square Root Properties
Square root properties play a pivotal role in understanding and finding the domain of expressions involving square roots. The square root function, denoted as \(\sqrt{x}\), is only defined for non-negative values of \(x\).
Some foundational properties of square roots include:\
These properties ensure that all operations stay within the real number system and are essential for simplifying and manipulating expressions with square roots. Understanding these properties helps us make sense of why the domain of \(\sqrt{147-3 x^{2}}\) is between -7 and 7 inclusive, and why we solve radicand inequalities the way we do.
Some foundational properties of square roots include:\
- \
- For any non-negative number \(a\), \(\sqrt{a^2} = a\). \
- For two non-negative numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) and \(\sqrt{a} / \sqrt{b} = \sqrt{a/b}\), given \(b eq 0\). \
- The square root of a non-negative number is also non-negative, which is why the radicand must be greater than or equal to zero. \
These properties ensure that all operations stay within the real number system and are essential for simplifying and manipulating expressions with square roots. Understanding these properties helps us make sense of why the domain of \(\sqrt{147-3 x^{2}}\) is between -7 and 7 inclusive, and why we solve radicand inequalities the way we do.
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