Find The Domain Of The Following Function. Express Your Answer In Interval Notation. Leave The Numerical Part Of Your Answer In Exact Form (i.e., No Decimals), If Needed.$ F(x) = \frac{\sqrt{14+x}}{5-x} }$Domain { \square$ $

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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. In other words, it is the set of all possible values of the variable (in this case, x) that the function can accept without resulting in an undefined or imaginary output. In this article, we will explore how to find the domain of a function, using the given function f(x) = √(14+x)/(5-x) as an example.

Understanding the Domain

The domain of a function can be restricted by various factors, including:

  • Denominator cannot be zero: The denominator of a fraction cannot be zero, as division by zero is undefined.
  • Radical expression: The expression inside a radical (such as a square root) must be non-negative.
  • Domain restrictions: Some functions may have domain restrictions imposed by the context in which they are used.

Step 1: Identify the Restrictions

To find the domain of the given function f(x) = √(14+x)/(5-x), we need to identify the restrictions imposed by the denominator and the radical expression.

Denominator Cannot Be Zero

The denominator of the function is 5-x. To find the values of x that make the denominator zero, we set 5-x = 0 and solve for x.

from sympy import symbols, Eq, solve

x = symbols('x')



equation = Eq(5 - x, 0)



solution = solve(equation, x)


print(solution)

The solution to the equation is x = 5. This means that the function is undefined when x = 5, as the denominator would be zero.

Radical Expression Must Be Non-Negative

The expression inside the radical is 14+x. To find the values of x that make this expression non-negative, we set 14+x ≥ 0 and solve for x.

from sympy import symbols, Eq, solve


x = symbols('x')



inequality = 14 + x >= 0



solution = solve(inequality, x)


print(solution)

The solution to the inequality is x ≥ -14. This means that the function is defined when x ≥ -14.

Step 2: Combine the Restrictions

Now that we have identified the restrictions imposed by the denominator and the radical expression, we can combine them to find the domain of the function.

The function is undefined when x = 5, as the denominator would be zero. Additionally, the function is defined when x ≥ -14, as the expression inside the radical would be non-negative.

To find the domain of the function, we need to combine these two restrictions. We can do this by finding the intersection of the two sets of values.

The set of values that make the denominator zero is {5}. The set of values that make the expression inside the radical non-negative is [-14, ∞).

The intersection of these two sets is the empty set, as there are no values that satisfy both restrictions.

However, we can see that the function is defined when x > -14 and x ≠ 5. This is because the function is defined when the expression inside the radical is non-negative, and it is undefined when the denominator is zero.

Therefore, the domain of the function is (-∞, -14) ∪ (-14, 5) ∪ (5, ∞).

Conclusion

In this article, we explored how to find the domain of a function using the given function f(x) = √(14+x)/(5-x) as an example. We identified the restrictions imposed by the denominator and the radical expression, and combined them to find the domain of the function.

The domain of the function is (-∞, -14) ∪ (-14, 5) ∪ (5, ∞). This means that the function is defined when x is less than -14, between -14 and 5 (exclusive), and greater than 5.

Introduction

In our previous article, we explored how to find the domain of a function using the given function f(x) = √(14+x)/(5-x) as an example. 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 for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, x) that the function can accept without resulting in an undefined or imaginary output.

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

A: To find the domain of a function, you need to identify the restrictions imposed by the denominator and the radical expression. The denominator cannot be zero, and the expression inside the radical must be non-negative. You can then combine these restrictions to find the domain of the function.

Q: What are some common restrictions that can limit the domain of a function?

A: Some common restrictions that can limit the domain of a function include:

  • Denominator cannot be zero: The denominator of a fraction cannot be zero, as division by zero is undefined.
  • Radical expression: The expression inside a radical (such as a square root) must be non-negative.
  • Domain restrictions: Some functions may have domain restrictions imposed by the context in which they are used.

Q: How do I determine if a function is defined at a particular value of x?

A: To determine if a function is defined at a particular value of x, you need to check if the denominator is zero and if the expression inside the radical is non-negative. If the denominator is zero, the function is undefined at that value of x. If the expression inside the radical is non-negative, the function is defined at that value of x.

Q: Can a function have multiple domains?

A: Yes, a function can have multiple domains. For example, a function may be defined for all real numbers except for a certain value, and then defined for all real numbers except for another certain value.

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): The function is defined for all real numbers less than a.
  • (a, b): The function is defined for all real numbers between a and b (exclusive).
  • (b, ∞): The function is defined for all real numbers greater than b.
  • (-∞, a) ∪ (b, ∞): The function is defined for all real numbers less than a and all real numbers greater than b.

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. This means that the function is undefined for all real numbers.

Conclusion

In this article, we answered some frequently asked questions about the domain of a function. We hope that this article has provided a clear and concise explanation of the domain of a function and how to find it. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

If you want to learn more about the domain of a function, we recommend checking out the following resources:

  • Khan Academy: Khan Academy has a comprehensive video series on the domain of a function.
  • Mathway: Mathway is an online math problem solver that can help you find the domain of a function.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you find the domain of a function.

We hope that this article has been helpful in understanding the domain of a function. If you have any questions or need further clarification, please don't hesitate to ask.