9. Find The Domain Of $y = 2 \sqrt{2x - 2}$.A. $x \ \textgreater \ 1$B. \$x \geq -1$[/tex\]C. $x \geq 1$D. $x \ \textgreater \ -1$

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Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with square root functions, it's essential to consider the restrictions on the domain caused by the presence of the square root symbol. In this article, we will explore the concept of the domain of a square root function, focusing on the specific case of the function $y = 2 \sqrt{2x - 2}$.

Understanding the Square Root Function

The square root function is defined as the inverse of the squaring function. It takes a non-negative real number as input and returns a non-negative real number as output. However, when dealing with the square root of a variable expression, we must ensure that the expression inside the square root is non-negative.

Restrictions on the Domain

For the function $y = 2 \sqrt{2x - 2}$, we need to find the values of x that make the expression inside the square root non-negative. This means that we must have:

2x−2≥02x - 2 \geq 0

Solving this inequality, we get:

2x≥22x \geq 2

Dividing both sides by 2, we obtain:

x≥1x \geq 1

Analyzing the Options

Now that we have found the domain of the function, let's analyze the given options:

A. $x \ \textgreater \ 1$

This option is incorrect because the domain of the function is $x \geq 1$, not $x \ \textgreater \ 1$.

B. $x≥−1x \geq -1[/tex]

This option is incorrect because the domain of the function is $x \geq 1$, not $x \geq -1$.

C. $x \geq 1$

This option is correct because it matches the domain of the function that we found earlier.

D. $x \ \textgreater \ -1$

This option is incorrect because the domain of the function is $x \geq 1$, not $x \ \textgreater \ -1$.

Conclusion

In conclusion, the domain of the function $y = 2 \sqrt{2x - 2}$ is $x \geq 1$. This means that the function is defined for all values of x greater than or equal to 1. We can verify this by plugging in values of x that satisfy this condition and checking that the function produces a real output.

Example Problems

To further reinforce our understanding of the domain of the square root function, let's consider a few example problems:

Example 1

Find the domain of the function $y = \sqrt{x^2 - 4}$.

Solution

To find the domain of this function, we need to ensure that the expression inside the square root is non-negative. This means that we must have:

x2−4≥0x^2 - 4 \geq 0

Solving this inequality, we get:

x2≥4x^2 \geq 4

Taking the square root of both sides, we obtain:

x≥2 or x≤−2x \geq 2 \text{ or } x \leq -2

Therefore, the domain of the function is $x \leq -2 \text{ or } x \geq 2$.

Example 2

Find the domain of the function $y = \sqrt{4 - x^2}$.

Solution

To find the domain of this function, we need to ensure that the expression inside the square root is non-negative. This means that we must have:

4−x2≥04 - x^2 \geq 0

Solving this inequality, we get:

x2≤4x^2 \leq 4

Taking the square root of both sides, we obtain:

−2≤x≤2-2 \leq x \leq 2

Therefore, the domain of the function is $-2 \leq x \leq 2$.

Final Thoughts

Q&A: 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 other words, it is the set of all values of the input variable 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 ensure that the expression inside the square root is non-negative. This means that you must have:

expression≥0expression \geq 0

Q: What if the expression inside the square root is negative? A: If the expression inside the square root is negative, then the function is not defined for that value of the input variable. In other words, the function has a "hole" or a "gap" in its domain.

Q: Can I have a negative value inside the square root? A: No, you cannot have a negative value inside the square root. The expression inside the square root must be non-negative, which means that it must be greater than or equal to zero.

Q: What if I have a fraction inside the square root? A: If you have a fraction inside the square root, you need to ensure that the numerator is non-negative and the denominator is positive. This is because a fraction is only defined when the numerator is non-negative and the denominator is positive.

Q: Can I have a square root of a negative number? A: No, you cannot have a square root of a negative number. The square root of a negative number is not a real number, and therefore it is not defined.

Q: What is the domain of the function $y = \sqrt{x^2 - 4}$? A: To find the domain of this function, we need to ensure that the expression inside the square root is non-negative. This means that we must have:

x2−4≥0x^2 - 4 \geq 0

Solving this inequality, we get:

x2≥4x^2 \geq 4

Taking the square root of both sides, we obtain:

x≥2 or x≤−2x \geq 2 \text{ or } x \leq -2

Therefore, the domain of the function is $x \leq -2 \text{ or } x \geq 2$.

Q: What is the domain of the function $y = \sqrt{4 - x^2}$? A: To find the domain of this function, we need to ensure that the expression inside the square root is non-negative. This means that we must have:

4−x2≥04 - x^2 \geq 0

Solving this inequality, we get:

x2≤4x^2 \leq 4

Taking the square root of both sides, we obtain:

−2≤x≤2-2 \leq x \leq 2

Therefore, the domain of the function is $-2 \leq x \leq 2$.

Q: How do I determine the domain of a square root function with multiple variables? A: To determine the domain of a square root function with multiple variables, you need to ensure that the expression inside the square root is non-negative for all values of the variables. This means that you must have:

expression≥0expression \geq 0

for all values of the variables.

Q: Can I have a square root of a variable expression? A: Yes, you can have a square root of a variable expression. However, you need to ensure that the expression inside the square root is non-negative for all values of the variable.

Q: What is the domain of the function $y = \sqrt{2x - 2}$? A: To find the domain of this function, we need to ensure that the expression inside the square root is non-negative. This means that we must have:

2x−2≥02x - 2 \geq 0

Solving this inequality, we get:

2x≥22x \geq 2

Dividing both sides by 2, we obtain:

x≥1x \geq 1

Therefore, the domain of the function is $x \geq 1$.

Conclusion

In conclusion, the domain of a square root function is determined by the values of the input variable that make the expression inside the square root non-negative. By understanding the restrictions on the domain, we can ensure that we are working with a well-defined function. In this article, we have explored the concept of the domain of a square root function, focusing on the specific case of the function $y = 2 \sqrt{2x - 2}$. We have also provided example problems and answered frequently asked questions to reinforce our understanding of the domain of the square root function.