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 , ∞ ],

Introduction

In mathematics, functions are used to describe the relationship between variables. 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 with base 3. The exponential function $3^x$ is defined for all real numbers $x$. This means that the function $f(x) = 3^x + 5$ is also defined for all real numbers $x$.

Determining the Domain

Since the function $f(x) = 3^x + 5$ is defined for all real numbers $x$, the domain of the function is the set of all real numbers. In interval notation, this can be written as $(-\infty, \infty)$.

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 with base 3. The exponential function $3^x$ is always positive, and when we add 5 to it, the result is always greater than 5. This means that the function $f(x) = 3^x + 5$ is always greater than 5.

Determining the Range

Since the function $f(x) = 3^x + 5$ is always greater than 5, the range of the function is the set of all real numbers greater than 5. In interval notation, this can be written as $(5, \infty)$.

Conclusion

In conclusion, the domain of the function $f(x) = 3^x + 5$ is the set of all real numbers, and the range of the function is the set of all real numbers greater than 5. This means that the correct answer is:

Domain: $(-\infty, \infty)$ Range: $(5, \infty)$

Final Answer

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 functions.

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 identify any restrictions on the input values. For example, if a function has a denominator of $x-2$, then $x$ cannot be equal to 2, because division by zero is undefined.

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 analyze the behavior of the function. For example, if a function is always increasing, then its range is all real numbers greater than or equal to the minimum value of the function.

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, the function $f(x) = x$ has a domain and range of all real numbers.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except 0.

Q: Can the range of a function be empty?

A: No, the range of a function cannot be empty. By definition, the range of a function is the set of all possible output values, and there is always at least one possible output value.

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

A: To graph the domain and range of a function, you can use a coordinate plane. The domain of the function is represented by the x-axis, and the range of the function is represented by the y-axis.

Q: Can I use technology to graph the domain and range of a function?

A: Yes, you can use technology such as graphing calculators or computer software to graph the domain and range of a function.

Conclusion

In conclusion, the domain and range of a function are important concepts in mathematics. By understanding the domain and range of a function, you can analyze its behavior and make predictions about its output values.

Final Tips

• Always check the domain and range of a function before graphing it.
• Use technology to graph the domain and range of a function if possible.
• Practice graphing the domain and range of functions to become more comfortable with the concepts.

Common Mistakes

• Failing to check the domain and range of a function before graphing it.
• Graphing a function with an empty domain or range.
• Failing to use technology to graph the domain and range of a function.

Additional Resources

• Khan Academy: Domain and Range of Functions
• Mathway: Domain and Range of Functions
• Wolfram Alpha: Domain and Range of Functions