What Are The Domain And Range Of The Function $f(x) = \log(x + 6) - 4$?A. Domain: $x \ \textgreater \ -6$, Range: $y \ \textgreater \ -4$B. Domain: $x \ \textgreater \ -6$, Range: All Real NumbersC. Domain:

Introduction

In mathematics, the domain and range of a function are crucial concepts that help us understand the behavior and properties of the function. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of the function $f(x) = \log(x + 6) - 4$.

What is the Domain of a Logarithmic Function?

The domain of a logarithmic function is the set of all possible input values (x-values) that the function can accept. For the function $f(x) = \log(x + 6) - 4$, we need to find the values of x that make the expression inside the logarithm positive.

The Logarithm Function: A Brief Review

The logarithm function is defined as $\log_b(x) = y \iff b^y = x$, where $b$ is the base of the logarithm. The logarithm function is only defined for positive real numbers, and the domain of the logarithm function is $(0, \infty)$.

Finding the Domain of the Function $f(x) = \log(x + 6) - 4$

To find the domain of the function $f(x) = \log(x + 6) - 4$, we need to find the values of x that make the expression inside the logarithm positive. The expression inside the logarithm is $x + 6$, and we need to find the values of x that make this expression positive.

Solving the Inequality $x + 6 > 0$

To find the values of x that make the expression inside the logarithm positive, we need to solve the inequality $x + 6 > 0$. Subtracting 6 from both sides of the inequality, we get $x > -6$.

Conclusion: The Domain of the Function $f(x) = \log(x + 6) - 4$

The domain of the function $f(x) = \log(x + 6) - 4$ is the set of all real numbers greater than -6. This means that the function is defined for all real numbers greater than -6.

What is the Range of a Logarithmic Function?

The range of a logarithmic function is the set of all possible output values (y-values) that the function can produce. For the function $f(x) = \log(x + 6) - 4$, we need to find the values of y that the function can produce.

The Logarithm Function: A Brief Review

The logarithm function is defined as $\log_b(x) = y \iff b^y = x$, where $b$ is the base of the logarithm. The logarithm function is only defined for positive real numbers, and the range of the logarithm function is all real numbers.

Finding the Range of the Function $f(x) = \log(x + 6) - 4$

To find the range of the function $f(x) = \log(x + 6) - 4$, we need to find the values of y that the function can produce. The function is a logarithmic function with a base of 10, and the expression inside the logarithm is $x + 6$. Since the expression inside the logarithm is always positive, the function will always produce positive values.

Solving the Inequality $y > -4$

To find the values of y that the function can produce, we need to solve the inequality $y > -4$. This inequality represents the set of all real numbers greater than -4.

Conclusion: The Range of the Function $f(x) = \log(x + 6) - 4$

The range of the function $f(x) = \log(x + 6) - 4$ is the set of all real numbers greater than -4. This means that the function can produce any real number greater than -4.

Conclusion

In conclusion, the domain of the function $f(x) = \log(x + 6) - 4$ is the set of all real numbers greater than -6, and the range of the function is the set of all real numbers greater than -4. This means that the function is defined for all real numbers greater than -6 and can produce any real number greater than -4.

Answer

The correct answer is:

A. Domain: $x \ \textgreater \ -6$, Range: $y \ \textgreater \ -4$

Discussion

The domain and range of a function are crucial concepts that help us understand the behavior and properties of the function. In this article, we explored the domain and range of the function $f(x) = \log(x + 6) - 4$. We found that the domain of the function is the set of all real numbers greater than -6, and the range of the function is the set of all real numbers greater than -4.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Domain and Range of a Function" by Khan Academy

Note

Frequently Asked Questions

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

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all possible input values (x-values) that the function can accept. For the function $f(x) = \log(x + 6) - 4$, the domain is the set of all real numbers greater than -6.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is the set of all possible output values (y-values) that the function can produce. For the function $f(x) = \log(x + 6) - 4$, the range is the set of all real numbers greater than -4.

Q: Why is the domain of a logarithmic function important?

A: The domain of a logarithmic function is important because it determines the values of x that the function can accept. If the domain is not specified, the function may not be defined for all possible values of x.

Q: Why is the range of a logarithmic function important?

A: The range of a logarithmic function is important because it determines the values of y that the function can produce. If the range is not specified, the function may not be defined for all possible values of y.

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

A: To find the domain and range of a logarithmic function, you need to follow these steps:

1. Identify the base of the logarithm.
2. Identify the expression inside the logarithm.
3. Determine the values of x that make the expression inside the logarithm positive.
4. Determine the values of y that the function can produce.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a logarithmic function have a domain of all real numbers?

A: No, a logarithmic function cannot have a domain of all real numbers. The domain of a logarithmic function is always a subset of the real numbers.

Q: Can a logarithmic function have a range of all real numbers?

A: Yes, a logarithmic function can have a range of all real numbers. For example, the function $f(x) = \log(x)$ has a range of all real numbers.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to follow these steps:

1. Identify the base of the logarithm.
2. Identify the expression inside the logarithm.
3. Determine the values of x that make the expression inside the logarithm positive.
4. Plot the points on a coordinate plane.
5. Draw a smooth curve through the points.

Q: What is the difference between a logarithmic function and an exponential function?

A: A logarithmic function is the inverse of an exponential function. While an exponential function grows rapidly, a logarithmic function grows slowly.

Q: Can a logarithmic function be used to model real-world phenomena?

A: Yes, a logarithmic function can be used to model real-world phenomena such as population growth, chemical reactions, and financial transactions.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Domain and Range of a Function" by Khan Academy
• [3] "Graphing Logarithmic Functions" by Mathway

Note

This article is for educational purposes only and is not intended to be a comprehensive review of the topic.