Find The Domain Of $y=\sqrt{3x+3}$.A. $x \geq 1$B. \$x \ \textgreater \ 1$[/tex\]C. $x \ \textgreater \ -1$D. $x \geq -1$
Introduction
When dealing with square root functions, it's essential to consider the domain of the function. The domain of a function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to values that make the expression inside the square root non-negative. In this article, we will explore how to find the domain of the function $y=\sqrt{3x+3}$.
Understanding Square Root Functions
A square root function is defined as $y=\sqrt{x}$, where $x$ is the input value. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4\times4=16$. However, not all numbers have a real square root. For instance, the square root of -1 is not a real number.
The Domain of a Square Root Function
The domain of a square root function is restricted to non-negative values. This means that the expression inside the square root must be greater than or equal to zero. In mathematical terms, the domain of a square root function can be represented as $x\geq0$.
Finding the Domain of $y=\sqrt{3x+3}$
To find the domain of the function $y=\sqrt{3x+3}$, we need to determine the values of $x$ that make the expression inside the square root non-negative. In this case, the expression inside the square root is $3x+3$. We can set up an inequality to represent this:
To solve this inequality, we can start by subtracting 3 from both sides:
Next, we can divide both sides by 3:
Simplifying the right-hand side, we get:
Therefore, the domain of the function $y=\sqrt{3x+3}$ is $x\geq-1$.
Conclusion
In conclusion, finding the domain of a square root function involves determining the values of the input variable that make the expression inside the square root non-negative. By setting up and solving an inequality, we can find the domain of the function. In the case of the function $y=\sqrt{3x+3}$, the domain is $x\geq-1$.
Answer
The correct answer is:
- D. $x \geq -1$
Introduction
In our previous article, we explored how to find the domain of a square root function. We used the function $y=\sqrt{3x+3}$ as an example and determined that the domain is $x\geq-1$. In this article, we will answer some frequently asked questions about finding the domain of a square root function.
Q: What is the domain of a square root function?
A: The domain of a square root function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to values that make the expression inside the square root non-negative.
Q: How do I find the domain of a square root function?
A: To find the domain of a square root function, you need to determine the values of the input variable that make the expression inside the square root non-negative. You can do this by setting up and solving an inequality.
Q: What is the inequality that I need to solve to find the domain of a square root function?
A: The inequality that you need to solve is $x\geq0$, where $x$ is the expression inside the square root.
Q: How do I solve the inequality $x\geq0$?
A: To solve the inequality $x\geq0$, you can start by subtracting 0 from both sides:
This simplifies to:
Q: What if the expression inside the square root is a linear expression, such as $3x+3$?
A: If the expression inside the square root is a linear expression, such as $3x+3$, you can set up an inequality to represent the condition that the expression must be non-negative. In this case, the inequality would be:
You can solve this inequality by subtracting 3 from both sides:
Next, you can divide both sides by 3:
Simplifying the right-hand side, you get:
Q: What if the expression inside the square root is a quadratic expression, such as $x^2+4x+4$?
A: If the expression inside the square root is a quadratic expression, such as $x^2+4x+4$, you can set up an inequality to represent the condition that the expression must be non-negative. In this case, the inequality would be:
You can solve this inequality by factoring the quadratic expression:
Since the square of any real number is non-negative, this inequality is always true. Therefore, the domain of the function is all real numbers.
Q: Can I use a calculator to find the domain of a square root function?
A: Yes, you can use a calculator to find the domain of a square root function. However, it's always a good idea to understand the underlying mathematics and to check your calculator's results.
Conclusion
In conclusion, finding the domain of a square root function involves determining the values of the input variable that make the expression inside the square root non-negative. By setting up and solving an inequality, you can find the domain of the function. We hope that this Q&A article has been helpful in answering your questions about finding the domain of a square root function.