Consider The Function $f(x)=\sqrt{5x-5}+1$.Which Inequality Is Used To Find The Domain?A. $5x-4 \geq 0$B. $\sqrt{5x-5}+1 \geq 0$C. $5x \geq 0$D. $5x-5 \geq 0$

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When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

In this article, we'll focus on finding the domain of the function f(x)=5x−5+1f(x)=\sqrt{5x-5}+1. To do this, we need to consider the inequality that is used to find the domain.

The Importance of Domain Inequality

The domain inequality is a mathematical expression that determines the set of all possible input values for a function. It's used to identify the values of x that make the function defined and real-valued.

For the given function f(x)=5x−5+1f(x)=\sqrt{5x-5}+1, we need to find the domain inequality. This inequality will help us determine the set of all possible x-values that the function can accept.

Finding the Domain Inequality

To find the domain inequality, we need to consider the expression inside the square root. The expression inside the square root must be non-negative, as the square root of a negative number is undefined.

Let's analyze the expression inside the square root: 5x−55x-5. This expression must be greater than or equal to zero for the function to be defined.

Solving the Inequality

To solve the inequality 5x−5≥05x-5 \geq 0, we need to isolate the variable x.

# Import necessary modules
import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the inequality
inequality = 5*x - 5 >= 0

# Solve the inequality
solution = sp.solve(inequality, x)

print(solution)

The solution to the inequality is x≥1x \geq 1.

Conclusion

In conclusion, the domain inequality for the function f(x)=5x−5+1f(x)=\sqrt{5x-5}+1 is 5x−5≥05x-5 \geq 0. This inequality determines the set of all possible input values for the function.

By solving the inequality, we found that the domain of the function is x≥1x \geq 1. This means that the function is defined for all x-values greater than or equal to 1.

Answer

The correct answer is D. 5x−5≥05x-5 \geq 0.

Additional Tips and Tricks

When dealing with functions, it's essential to understand the concept of the domain. The domain inequality is a mathematical expression that determines the set of all possible input values for a function.

Here are some additional tips and tricks to help you find the domain inequality:

  • Always consider the expression inside the square root. The expression inside the square root must be non-negative for the function to be defined.
  • Use algebraic manipulations to isolate the variable x.
  • Solve the inequality using mathematical software or online tools.
  • Verify the solution by plugging in test values.

By following these tips and tricks, you'll be able to find the domain inequality for any given function.

Common Mistakes to Avoid

When dealing with functions, it's essential to avoid common mistakes. Here are some common mistakes to avoid:

  • Not considering the expression inside the square root.
  • Not isolating the variable x.
  • Not solving the inequality correctly.
  • Not verifying the solution.

By avoiding these common mistakes, you'll be able to find the domain inequality for any given function.

Conclusion

In conclusion, the domain inequality for the function f(x)=5x−5+1f(x)=\sqrt{5x-5}+1 is 5x−5≥05x-5 \geq 0. This inequality determines the set of all possible input values for the function.

By solving the inequality, we found that the domain of the function is x≥1x \geq 1. This means that the function is defined for all x-values greater than or equal to 1.

In our previous article, we discussed the concept of the domain inequality and how to find it for the function f(x)=5x−5+1f(x)=\sqrt{5x-5}+1. However, we know that there are many more questions and concerns that you may have. In this article, we'll address some of the most frequently asked questions about domain inequality.

Q: What is the domain inequality?

A: The domain inequality is a mathematical expression that determines the set of all possible input values for a function. It's used to identify the values of x that make the function defined and real-valued.

Q: Why is the domain inequality important?

A: The domain inequality is important because it helps us understand the behavior of a function. By knowing the domain inequality, we can determine the set of all possible input values for a function, which is essential for solving equations and inequalities.

Q: How do I find the domain inequality?

A: To find the domain inequality, you need to consider the expression inside the square root. The expression inside the square root must be non-negative for the function to be defined. You can use algebraic manipulations to isolate the variable x and solve the inequality.

Q: What are some common mistakes to avoid when finding the domain inequality?

A: Some common mistakes to avoid when finding the domain inequality include:

  • Not considering the expression inside the square root.
  • Not isolating the variable x.
  • Not solving the inequality correctly.
  • Not verifying the solution.

Q: How do I verify the solution to the domain inequality?

A: To verify the solution to the domain inequality, you can plug in test values into the function and check if the output is defined and real-valued. You can also use mathematical software or online tools to verify the solution.

Q: What are some real-world applications of the domain inequality?

A: The domain inequality has many real-world applications, including:

  • Physics: The domain inequality is used to determine the range of values for which a physical system is valid.
  • Engineering: The domain inequality is used to determine the range of values for which a mechanical system is valid.
  • Economics: The domain inequality is used to determine the range of values for which a economic model is valid.

Q: Can I use the domain inequality to solve equations and inequalities?

A: Yes, you can use the domain inequality to solve equations and inequalities. By knowing the domain inequality, you can determine the set of all possible input values for a function, which is essential for solving equations and inequalities.

Q: How do I use the domain inequality to solve equations and inequalities?

A: To use the domain inequality to solve equations and inequalities, you need to:

  • Identify the function and the equation or inequality to be solved.
  • Determine the domain inequality for the function.
  • Use the domain inequality to determine the set of all possible input values for the function.
  • Solve the equation or inequality using the domain inequality.

Q: What are some tips and tricks for finding the domain inequality?

A: Some tips and tricks for finding the domain inequality include:

  • Always consider the expression inside the square root. The expression inside the square root must be non-negative for the function to be defined.
  • Use algebraic manipulations to isolate the variable x.
  • Solve the inequality using mathematical software or online tools.
  • Verify the solution by plugging in test values.

By following these tips and tricks, you'll be able to find the domain inequality for any given function.

Conclusion

In conclusion, the domain inequality is a mathematical expression that determines the set of all possible input values for a function. By knowing the domain inequality, you can determine the set of all possible input values for a function, which is essential for solving equations and inequalities.

We hope this article has helped you understand the concept of the domain inequality and how to find it for any given function. If you have any further questions or concerns, please don't hesitate to ask.