What Are The Domain And Range Of The Function F ( X ) = 3 X + 5 F(x) = 3^x + 5 F ( X ) = 3 X + 5 ?A. Domain: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]; Range: ( 0 , ∞ (0, \infty ( 0 , ∞ ]B. Domain: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]; Range: ( 5 , ∞ (5, \infty ( 5 , ∞ ]C. Domain: ( 0 , ∞ (0, \infty ( 0 , ∞ ];

Mar 13, 2025 by ADMIN 311 views

**Understanding the Domain and Range of a Function**

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will explore the domain and range of the function $f(x) = 3^x + 5$ .

What is the Domain of a Function?

The domain of a function is the set of all possible input values. In other words, it is the set of all values of $x$ for which the function is defined. For the function $f(x) = 3^x + 5$ , we need to determine the set of all possible values of $x$ .

Analyzing the Function

The function $f(x) = 3^x + 5$ is an exponential function, where the base is 3 and the exponent is $x$ . The function is defined for all real numbers, which means that the domain of the function is all real numbers, denoted by $(-\infty, \infty)$ .

Why is the Domain All Real Numbers?

The function $f(x) = 3^x + 5$ is defined for all real numbers because the base, 3, is a positive number, and the exponent, $x$ , is a real number. The exponential function $3^x$ is defined for all real numbers, and adding 5 to the result does not change the domain.

What is the Range of a Function?

The range of a function is the set of all possible output values. In other words, it is the set of all values of $f(x)$ for which the function is defined. For the function $f(x) = 3^x + 5$ , we need to determine the set of all possible values of $f(x)$ .

Analyzing the Function

The function $f(x) = 3^x + 5$ is an exponential function, where the base is 3 and the exponent is $x$ . The function is defined for all real numbers, and the minimum value of the function is 5, which occurs when $x = -\infty$ . As $x$ increases, the value of $f(x)$ also increases without bound.

Why is the Range All Positive Numbers Greater than 5?

The range of the function $f(x) = 3^x + 5$ is all positive numbers greater than 5 because the minimum value of the function is 5, and the function increases without bound as $x$ increases. The base, 3, is a positive number, and the exponent, $x$ , is a real number, so the function is always positive.

Conclusion

In conclusion, the domain of the function $f(x) = 3^x + 5$ is all real numbers, denoted by $(-\infty, \infty)$ , and the range is all positive numbers greater than 5, denoted by $(5, \infty)$ . This is because the function is defined for all real numbers, and the minimum value of the function is 5, which occurs when $x = -\infty$ .

Answer

The correct answer is B. Domain: $(-\infty, \infty)$ ; Range: $(5, \infty)$ .

References

[1] "Functions" by Khan Academy
[2] "Exponential Functions" by Math Is Fun
[3] "Domain and Range" by Purplemath
Domain and Range of a Function: Q&A =====================================

Introduction

In our previous article, we explored the domain and range of the function $f(x) = 3^x + 5$ . In this article, we will answer some frequently asked questions about the domain and range 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. In other words, it is the set of all values of $x$ 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 consider the values of $x$ for which the function is defined. For example, if the function has a denominator, you need to make sure that the denominator is not equal to zero.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. In other words, it is the set of all values of $f(x)$ for which the function is defined.

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

A: To determine the range of a function, you need to consider the values of $f(x)$ for which the function is defined. For example, if the function has a minimum or maximum value, you need to consider that value when determining the range.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, if the function is a constant function, the domain and range are both the same set of values.

Q: Can the domain and range of a function be different?

A: Yes, the domain and range of a function can be different. For example, if the function is a linear function, the domain and range may be different sets of values.

Q: How do I graph a function with a given domain and range?

A: To graph a function with a given domain and range, you need to consider the values of $x$ and $f(x)$ that are within the domain and range. You can use a graphing calculator or software to help you graph the function.

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

A: The domain and range of a function are important because they help you understand the behavior of the function. For example, if the domain of a function is all real numbers, the function is defined for all possible input values. If the range of a function is all positive numbers, the function always produces positive output values.

Q: Can the domain and range of a function change?

A: Yes, the domain and range of a function can change. For example, if the function is a piecewise function, the domain and range may be different for different parts of the function.

Q: How do I determine the domain and range of a function with multiple variables?

A: To determine the domain and range of a function with multiple variables, you need to consider the values of all the variables that are within the domain and range. You can use techniques such as substitution and elimination to help you determine the domain and range.

Conclusion

In conclusion, the domain and range of a function are important concepts that help you understand the behavior of the function. By considering the values of $x$ and $f(x)$ that are within the domain and range, you can graph the function and understand its behavior.

References

[1] "Functions" by Khan Academy
[2] "Exponential Functions" by Math Is Fun
[3] "Domain and Range" by Purplemath