Find The Domain Of The Following Function. F ( X ) = 2 X − 16 F(x) = \sqrt{2x - 16} F ( X ) = 2 X − 16 ​ Give Your Answer In Interval Notation.

Introduction

When dealing with functions that involve square roots, 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 expression inside the square root must be non-negative. In this article, we will explore how to find the domain of the function $f(x) = \sqrt{2x - 16}$.

Understanding Square Root Functions

A square root function is defined as $f(x) = \sqrt{x}$, 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 $4 \times 4 = 16$. However, the square root of a negative number is not a real number, as there is no real number that can be multiplied by itself to give a negative value.

The Domain of a Square Root Function

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 the function $f(x) = \sqrt{x}$, the domain is all real numbers greater than or equal to 0, denoted as $[0, \infty)$. This is because the expression inside the square root must be non-negative.

Finding the Domain of the Function $f(x) = \sqrt{2x - 16}$

To find the domain of the function $f(x) = \sqrt{2x - 16}$, we need to consider the expression inside the square root, which is $2x - 16$. This expression must be non-negative, as the square root of a negative number is not a real number.

Step 1: Set Up the Inequality

To find the domain of the function, we need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. In this case, the inequality is:

$$2x - 16 \geq 0$$

Step 2: Solve the Inequality

To solve the inequality, we need to isolate the variable $x$. We can do this by adding 16 to both sides of the inequality, which gives us:

$$2x \geq 16$$

Next, we can divide both sides of the inequality by 2, which gives us:

$$x \geq 8$$

Step 3: Write the Domain in Interval Notation

The domain of the function $f(x) = \sqrt{2x - 16}$ is all real numbers greater than or equal to 8. We can write this in interval notation as $[8, \infty)$.

Conclusion

In conclusion, the domain of the function $f(x) = \sqrt{2x - 16}$ is all real numbers greater than or equal to 8, denoted as $[8, \infty)$. This is because the expression inside the square root must be non-negative, and the inequality $x \geq 8$ represents this condition.

Example Problems

Problem 1

Find the domain of the function $f(x) = \sqrt{3x - 12}$.

Solution

To find the domain of the function, we need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. In this case, the inequality is:

$$3x - 12 \geq 0$$

Solving the inequality, we get:

$$x \geq 4$$

The domain of the function is all real numbers greater than or equal to 4, denoted as $[4, \infty)$.

Problem 2

Find the domain of the function $f(x) = \sqrt{x + 2}$.

Solution

To find the domain of the function, we need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. In this case, the inequality is:

$$x + 2 \geq 0$$

Solving the inequality, we get:

$$x \geq -2$$

The domain of the function is all real numbers greater than or equal to -2, denoted as $[-2, \infty)$.

Final Answer

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Introduction

In our previous article, we discussed how to find the domain of a square root 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 expression inside the square root must be non-negative. In this article, we will answer some frequently asked questions about 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 expression inside the square root must be 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 set up an inequality that represents the condition that the expression inside the square root must be non-negative. You can then solve the inequality to find the domain of the function.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values of the function. For example, the domain of the function $f(x) = \sqrt{x}$ is all real numbers greater than or equal to 0, while the range is all real numbers greater than or equal to 0.

Q: Can the domain of a function be a single value?

A: Yes, the domain of a function can be a single value. For example, the function $f(x) = \sqrt{x}$ has a domain of 0, because the expression inside the square root must be non-negative, and 0 is the only value that satisfies this condition.

Q: Can the domain of a function be an interval?

A: Yes, the domain of a function can be an interval. For example, the function $f(x) = \sqrt{x}$ has a domain of $[0, \infty)$, because the expression inside the square root must be non-negative, and all values greater than or equal to 0 satisfy this condition.

Q: How do I write the domain of a function in interval notation?

A: To write the domain of a function in interval notation, you need to use the following notation:

• $[a, b]$ represents the interval from $a$ to $b$, inclusive.
• $(a, b)$ represents the interval from $a$ to $b$, exclusive.
• $[a, \infty)$ represents the interval from $a$ to infinity, inclusive.
• $(-\infty, a)$ represents the interval from negative infinity to $a$, inclusive.

Q: Can the domain of a function be a union of intervals?

A: Yes, the domain of a function can be a union of intervals. For example, the function $f(x) = \sqrt{x}$ has a domain of $[0, \infty) \cup (-\infty, 0)$, because the expression inside the square root must be non-negative, and all values greater than or equal to 0 or less than 0 satisfy this condition.

Q: How do I find the domain of a composite function?

A: To find the domain of a composite function, you need to find the domain of each individual function and then find the intersection of the two domains. For example, if we have the composite function $f(g(x))$, we need to find the domain of $f(x)$ and the domain of $g(x)$, and then find the intersection of the two domains.

Conclusion

In conclusion, the domain of a square root function is the set of all possible input values for which the function is defined. To find the domain of a square root function, you need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. You can then solve the inequality to find the domain of the function. We hope this article has helped you understand the concept of the domain of a square root function and how to find it.

Example Problems

Problem 1

Find the domain of the function $f(x) = \sqrt{x + 2}$.

Solution

To find the domain of the function, we need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. In this case, the inequality is:

$$x + 2 \geq 0$$

Solving the inequality, we get:

$$x \geq -2$$

The domain of the function is all real numbers greater than or equal to -2, denoted as $[-2, \infty)$.

Problem 2

Find the domain of the function $f(x) = \sqrt{2x - 12}$.

Solution

To find the domain of the function, we need to set up an inequality that represents the condition that the expression inside the square root must be non-negative. In this case, the inequality is:

$$2x - 12 \geq 0$$

Solving the inequality, we get:

$$x \geq 6$$

The domain of the function is all real numbers greater than or equal to 6, denoted as $[6, \infty)$.

Final Answer

The final answer is $\boxed{[8, \infty)}$.