Select The Correct Answer.What Is The Domain Of The Function $f(x) = X^2 + 3x + 5$?A. All Whole Numbers B. All Positive Real Numbers C. All Integers D. All Real Numbers

Feb 27, 2025 by ADMIN 172 views

**Understanding the Domain of a Function**

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 real numbers that can be plugged into the function without causing any issues.

What is the Domain of a Function?

The domain of a function can be determined by looking at the function's equation. In the case of the function $f(x) = x^2 + 3x + 5$ , we need to consider the possible values of x that will not cause any problems when plugged into the function.

Analyzing the Function

Let's take a closer look at the function $f(x) = x^2 + 3x + 5$ . This is a quadratic function, which means it's a polynomial of degree 2. The general form of a quadratic function is $f(x) = ax^2 + bx + c$ , where a, b, and c are constants.

In this case, the function $f(x) = x^2 + 3x + 5$ has a leading coefficient of 1, which means it's a monic quadratic function. This means that the function will always have a positive leading coefficient, which will ensure that the function is always defined for all real numbers.

Determining the Domain

Since the function $f(x) = x^2 + 3x + 5$ is a quadratic function with a positive leading coefficient, it's defined for all real numbers. This means that the domain of the function is the set of all real numbers.

Why is the Domain All Real Numbers?

The reason why the domain of the function $f(x) = x^2 + 3x + 5$ is all real numbers is because the function is a quadratic function with a positive leading coefficient. This means that the function will always have a positive value for any real number input.

In other words, no matter what real number you plug into the function, you'll always get a positive value as output. This is because the square of any real number is always non-negative, and the sum of a non-negative number and a positive number is always positive.

Conclusion

In conclusion, the domain of the function $f(x) = x^2 + 3x + 5$ is all real numbers. This is because the function is a quadratic function with a positive leading coefficient, which means it's defined for all real numbers.

Answer

The correct answer is D. all real numbers.

Why is this the correct answer?

This is the correct answer because the function $f(x) = x^2 + 3x + 5$ is a quadratic function with a positive leading coefficient, which means it's defined for all real numbers.

What are the other options?

The other options are:

A. all whole numbers: This is not the correct answer because the function is not defined for all whole numbers. In fact, the function is defined for all real numbers, not just whole numbers.
B. all positive real numbers: This is not the correct answer because the function is not defined only for positive real numbers. In fact, the function is defined for all real numbers, not just positive real numbers.
C. all integers: This is not the correct answer because the function is not defined only for integers. In fact, the function is defined for all real numbers, not just integers.

What is the significance of the domain?

The domain of a function is significant because it tells us what values of x are allowed as input. In other words, it tells us what values of x will not cause any problems when plugged into the function.

How do we determine the domain?

We determine the domain by looking at the function's equation and considering the possible values of x that will not cause any problems when plugged into the function.

What are some common domains?

Some common domains include:

All real numbers
All positive real numbers
All negative real numbers
All integers
All whole numbers

What are some common functions with specific domains?

Some common functions with specific domains include:

The function $f(x) = 1/x$ has a domain of all non-zero real numbers.
The function $f(x) = \sqrt{x}$ has a domain of all non-negative real numbers.
The function $f(x) = 1/x^2$ has a domain of all non-zero real numbers.

What are some common mistakes when determining the domain?

Some common mistakes when determining the domain include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

How do we avoid these mistakes?

We avoid these mistakes by carefully analyzing the function's equation and considering the possible values of x that will not cause any problems when plugged into the function.

What are some tips for determining the domain?

Some tips for determining the domain include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

What are some common functions with specific domains in real-life applications?

Some common functions with specific domains in real-life applications include:

The function $f(x) = 1/x$ is used in physics to model the behavior of electric circuits.
The function $f(x) = \sqrt{x}$ is used in engineering to model the behavior of stress and strain in materials.
The function $f(x) = 1/x^2$ is used in economics to model the behavior of supply and demand curves.

What are some common mistakes when using functions with specific domains in real-life applications?

Some common mistakes when using functions with specific domains in real-life applications include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

How do we avoid these mistakes?

We avoid these mistakes by carefully analyzing the function's equation and considering the possible values of x that will not cause any problems when plugged into the function.

What are some tips for using functions with specific domains in real-life applications?

Some tips for using functions with specific domains in real-life applications include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

What are some common functions with specific domains in mathematics?

Some common functions with specific domains in mathematics include:

The function $f(x) = 1/x$ has a domain of all non-zero real numbers.
The function $f(x) = \sqrt{x}$ has a domain of all non-negative real numbers.
The function $f(x) = 1/x^2$ has a domain of all non-zero real numbers.

What are some common mistakes when using functions with specific domains in mathematics?

Some common mistakes when using functions with specific domains in mathematics include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

How do we avoid these mistakes?

We avoid these mistakes by carefully analyzing the function's equation and considering the possible values of x that will not cause any problems when plugged into the function.

What are some tips for using functions with specific domains in mathematics?

Some tips for using functions with specific domains in mathematics include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

What are some common functions with specific domains in science?

Some common functions with specific domains in science include:

The function $f(x) = 1/x$ is used in physics to model the behavior of electric circuits.
The function $f(x) = \sqrt{x}$ is used in engineering to model the behavior of stress and strain in materials.
The function $f(x) = 1/x^2$ is used in economics to model the behavior of supply and demand curves.

What are some common mistakes when using functions with specific domains in science?

Some common mistakes when using functions with specific domains in science include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

How do we avoid these mistakes?

We avoid these mistakes by carefully analyzing the function's equation and considering the possible values of x that will not cause any problems when plugged into the function.

**What are some tips for using functions with

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.

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

A: To determine the domain of a function, you need to look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Q: What are some common mistakes when determining the domain of a function?

A: Some common mistakes when determining the domain of a function include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

Q: How do I avoid these mistakes?

A: To avoid these mistakes, you need to carefully analyze the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Q: What are some tips for determining the domain of a function?

A: Some tips for determining the domain of a function include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

Q: What are some common functions with specific domains?

A: Some common functions with specific domains include:

The function $f(x) = 1/x$ has a domain of all non-zero real numbers.
The function $f(x) = \sqrt{x}$ has a domain of all non-negative real numbers.
The function $f(x) = 1/x^2$ has a domain of all non-zero real numbers.

Q: What are some common mistakes when using functions with specific domains?

A: Some common mistakes when using functions with specific domains include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

Q: How do I avoid these mistakes?

A: To avoid these mistakes, you need to carefully analyze the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Q: What are some tips for using functions with specific domains?

A: Some tips for using functions with specific domains include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

Q: What are some common functions with specific domains in real-life applications?

A: Some common functions with specific domains in real-life applications include:

The function $f(x) = 1/x$ is used in physics to model the behavior of electric circuits.
The function $f(x) = \sqrt{x}$ is used in engineering to model the behavior of stress and strain in materials.
The function $f(x) = 1/x^2$ is used in economics to model the behavior of supply and demand curves.

Q: What are some common mistakes when using functions with specific domains in real-life applications?

A: Some common mistakes when using functions with specific domains in real-life applications include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

Q: How do I avoid these mistakes?

A: To avoid these mistakes, you need to carefully analyze the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Q: What are some tips for using functions with specific domains in real-life applications?

A: Some tips for using functions with specific domains in real-life applications include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

Q: What are some common functions with specific domains in mathematics?

A: Some common functions with specific domains in mathematics include:

The function $f(x) = 1/x$ has a domain of all non-zero real numbers.
The function $f(x) = \sqrt{x}$ has a domain of all non-negative real numbers.
The function $f(x) = 1/x^2$ has a domain of all non-zero real numbers.

Q: What are some common mistakes when using functions with specific domains in mathematics?

A: Some common mistakes when using functions with specific domains in mathematics include:

Assuming that the function is defined for all real numbers when it's not.
Assuming that the function is defined only for positive real numbers when it's not.
Assuming that the function is defined only for integers when it's not.

Q: How do I avoid these mistakes?

A: To avoid these mistakes, you need to carefully analyze the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Q: What are some tips for using functions with specific domains in mathematics?

A: Some tips for using functions with specific domains in mathematics include:

Look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.
Consider the possible values of x that will cause the function to be undefined.
Consider the possible values of x that will cause the function to be infinite.
Consider the possible values of x that will cause the function to be negative.

Conclusion

In conclusion, the domain of a function is the set of all possible input values (x-values) for which the function is defined. To determine the domain of a function, you need to look at the function's equation and consider the possible values of x that will not cause any problems when plugged into the function. Some common functions with specific domains include the function $f(x) = 1/x$ , the function $f(x) = \sqrt{x}$ , and the function $f(x) = 1/x^2$ . Some common mistakes when using functions with specific domains include assuming that the function is defined for all real numbers when it's not, assuming that the function is defined only for positive real numbers when it's not, and assuming that the function is defined only for integers when it's not. To avoid these mistakes, you need to carefully analyze the function's equation and consider the possible values of x that will not cause any problems when plugged into the function.

Final Answer

The final answer is D. all real numbers.