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

Introduction

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 explore how to find the domain of a function, using the given function $f(x)=\sqrt{5x-5}+1$ as an example.

The Importance of Domain

The domain of a function is crucial in mathematics, as it determines the range of possible input values that the function can accept. If the domain is not properly defined, it can lead to incorrect results or even undefined outputs. For instance, if we try to find the square root of a negative number, the function will result in an imaginary output, which is not a real number.

Finding the Domain of a Function

To find the domain of a function, we need to identify the values of x that make the function undefined or imaginary. In the case of the given function $f(x)=\sqrt{5x-5}+1$, we need to find the values of x that make the expression inside the square root non-negative.

Step 1: Identify the Expression Inside the Square Root

The expression inside the square root is $5x-5$. To find the values of x that make this expression non-negative, we need to set it greater than or equal to zero.

Step 2: Solve the Inequality

We need to solve the inequality $5x-5 \geq 0$. To do this, we can add 5 to both sides of the inequality, resulting in $5x \geq 5$. Then, we can divide both sides by 5, resulting in $x \geq 1$.

Conclusion

In conclusion, the inequality used to find the domain of the function $f(x)=\sqrt{5x-5}+1$ is $x \geq 1$. This means that the function is defined for all values of x greater than or equal to 1.

Answer

The correct answer is:

• D. $5x-5 \geq 0$

This is the correct inequality used to find the domain of the function $f(x)=\sqrt{5x-5}+1$.

Example Use Case

Suppose we want to find the domain of the function $f(x)=\sqrt{x^2-4}+2$. To do this, we need to find the values of x that make the expression inside the square root non-negative. We can set up the inequality $x^2-4 \geq 0$ and solve for x. This will give us the domain of the function.

Tips and Tricks

When finding the domain of a function, it's essential to remember the following tips and tricks:

• Always identify the expression inside the square root and set it greater than or equal to zero.
• Solve the resulting inequality to find the values of x that make the function defined.
• Use the correct inequality to find the domain of the function.

By following these tips and tricks, you'll be able to find the domain of any function with ease.

Conclusion

Q: What is the domain of a function?

A: 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.

Q: Why is the domain of a function important?

A: The domain of a function is crucial in mathematics, as it determines the range of possible input values that the function can accept. If the domain is not properly defined, it can lead to incorrect results or even undefined outputs.

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

A: To find the domain of a function, you need to identify the values of x that make the function undefined or imaginary. In the case of a square root function, you need to find the values of x that make the expression inside the square root non-negative.

Q: What is the expression inside the square root?

A: The expression inside the square root is the part of the function that is under the square root symbol. For example, in the function $f(x)=\sqrt{5x-5}+1$, the expression inside the square root is $5x-5$.

Q: How do I solve the inequality to find the domain?

A: To solve the inequality, you need to set the expression inside the square root greater than or equal to zero. Then, you can solve for x by adding or subtracting the same value to both sides of the inequality, and then dividing both sides by the same non-zero value.

Q: What are some common mistakes to avoid when finding the domain of a function?

A: Some common mistakes to avoid when finding the domain of a function include:

• Not identifying the expression inside the square root
• Not setting the expression inside the square root greater than or equal to zero
• Not solving the inequality correctly
• Not considering the domain of the function when graphing or evaluating the function

Q: How do I graph a function with a domain restriction?

A: To graph a function with a domain restriction, you need to identify the domain of the function and then graph the function only for the values of x that are in the domain.

Q: Can I have a function with an empty domain?

A: Yes, it is possible to have a function with an empty domain. This occurs when the expression inside the square root is always negative, or when the function is undefined for all values of x.

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

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

A: To find the range of a function, you need to identify the set of all possible output values (y-values) that the function can produce. This can be done by evaluating the function for different values of x and identifying the resulting y-values.

Conclusion

In conclusion, finding the domain of a function is a crucial step in mathematics. By identifying the expression inside the square root and solving the resulting inequality, we can find the values of x that make the function defined. In this article, we answered some frequently asked questions about the domain of a function and provided tips and tricks for finding the domain of any function.