State The Domain Of The Function.$\[ F(x) = \sqrt{16-x} \\]

The domain of a function is the set of all possible input values (x) for which the function is defined. In other words, it is the set of all possible values of x that the function can accept without resulting in an undefined or imaginary output. In this article, we will explore the concept of the domain of a function, with a focus on the square root function.

What is the Domain of a Function?

The domain of a function is a critical concept in mathematics, particularly in algebra and calculus. It is essential to understand the domain of a function to ensure that we are working with valid and meaningful input values. The domain of a function can be thought of as the set of all possible values of x that the function can accept without resulting in an undefined or imaginary output.

Square Root Function: A Special Case

The square root function is a special case of a function that has a specific domain. The square root function is defined as:

${ f(x) = \sqrt{x} }$

The domain of the square root function is all non-negative real numbers, or x ≥ 0. This means that the square root function is only defined for input values of x that are greater than or equal to 0.

Domain of the Square Root Function: 16 - x

In the given function:

${ f(x) = \sqrt{16-x} }$

We need to find the domain of this function. To do this, we need to determine the values of x for which the expression inside the square root is non-negative.

Step 1: Determine the Values of x for Which the Expression Inside the Square Root is Non-Negative

The expression inside the square root is 16 - x. We need to find the values of x for which this expression is non-negative.

To do this, we can set up an inequality:

16 - x ≥ 0

Solving for x, we get:

x ≤ 16

This means that the expression inside the square root is non-negative for all values of x that are less than or equal to 16.

Step 2: Determine the Domain of the Function

Now that we have determined the values of x for which the expression inside the square root is non-negative, we can determine the domain of the function.

The domain of the function is all values of x that are less than or equal to 16. In interval notation, this can be written as:

[−∞, 16]

Conclusion

In conclusion, the domain of the function:

${ f(x) = \sqrt{16-x} }$

is all values of x that are less than or equal to 16. This can be written in interval notation as:

[−∞, 16]

The domain of a function is a critical concept in mathematics, and understanding the domain of a function is essential to ensure that we are working with valid and meaningful input values.

Key Takeaways

• The domain of a function is the set of all possible input values (x) for which the function is defined.
• The domain of a function can be thought of as the set of all possible values of x that the function can accept without resulting in an undefined or imaginary output.
• The square root function has a specific domain, which is all non-negative real numbers.
• To determine the domain of a function, we need to determine the values of x for which the expression inside the square root is non-negative.
• The domain of the function:

${ f(x) = \sqrt{16-x} }$

is all values of x that are less than or equal to 16.

Frequently Asked Questions

• What is the domain of a function?
• How do we determine the domain of a function?
• What is the domain of the square root function?
• How do we find the domain of a function with a square root?

References

• [1] "Domain of a Function" by Math Open Reference
• [2] "Square Root Function" by Wolfram MathWorld
• [3] "Domain of a Function with a Square Root" by Khan Academy
  Domain of a Function: Frequently Asked Questions =====================================================

In our previous article, we discussed the concept of the domain of a function, with a focus on the square root function. We also explored the domain of the function:

${ f(x) = \sqrt{16-x} }$

In this article, we will answer some frequently asked questions about the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x) for which the function is defined. In other words, it is the set of all possible values of x that the function can accept without resulting in an undefined or imaginary output.

Q: How do we determine the domain of a function?

A: To determine the domain of a function, we need to examine the function and identify any restrictions on the input values. We can do this by looking for any values of x that would result in an undefined or imaginary output.

Q: What is the domain of the square root function?

A: The domain of the square root function is all non-negative real numbers, or x ≥ 0. This means that the square root function is only defined for input values of x that are greater than or equal to 0.

Q: How do we find the domain of a function with a square root?

A: To find the domain of a function with a square root, we need to determine the values of x for which the expression inside the square root is non-negative. We can do this by setting up an inequality and solving for x.

Q: What is the domain of the function: f(x) = √(16-x)?

A: The domain of the function:

${ f(x) = \sqrt{16-x} }$

is all values of x that are less than or equal to 16. In interval notation, this can be written as:

[−∞, 16]

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:

${ f(x) = \frac{1}{x} }$

has a domain of x ≠ 0, which is a single value.

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 x ≥ 0, which is an interval.

Q: How do we represent the domain of a function?

A: We can represent the domain of a function using interval notation, which is a way of writing intervals using square brackets and parentheses. For example, the domain of the function:

${ f(x) = \sqrt{16-x} }$

can be written as:

[−∞, 16]

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 (x) for which the function is defined, while the range of a function is the set of all possible output values (y) that the function can produce.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, the function:

${ f(x) = x }$

has a domain and range of all real numbers.

Q: How do we find the domain and range of a function?

A: To find the domain and range of a function, we need to examine the function and identify any restrictions on the input and output values. We can do this by looking for any values of x that would result in an undefined or imaginary output, and by examining the graph of the function.

Conclusion

In conclusion, the domain of a function is a critical concept in mathematics, and understanding the domain of a function is essential to ensure that we are working with valid and meaningful input values. We hope that this article has helped to answer some of the frequently asked questions about the domain of a function.