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: { (-∞, ∞)$}$; Range: { (0, ∞)$}$ B. Domain: { (-∞, ∞)$}$; Range: { (5, ∞)$}$ C. Domain: { (0, ∞)$} ; R A N G E : \[ ; Range: \[ ; R An G E : \[ (-∞,

Mar 12, 2025 by ADMIN 256 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 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 $x$ . Therefore, the domain of the function is the set of all real numbers, which can be represented as $(-∞, ∞)$ .

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. Therefore, the range of the function is the set of all real numbers greater than 5, which can be represented as $(5, ∞)$ .

Conclusion

In conclusion, the domain of the function $f(x)=3^x+5$ is the set of all real numbers, which can be represented as $(-∞, ∞)$ . The range of the function is the set of all real numbers greater than 5, which can be represented as $(5, ∞)$ . Therefore, the correct answer is:

Domain: $(-∞, ∞)$ Range: $(5, ∞)$

Answer Key

A. Domain: $(-∞, ∞)$ ; Range: $(0, ∞)$ B. Domain: $(-∞, ∞)$ ; Range: $(5, ∞)$ C. Domain: $(0, ∞)$ ; Range: $(-∞, ∞)$

Introduction

In our previous article, we discussed 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 difference between the domain and range of a function?

A: 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 other words, the domain is the set of all values of $x$ for which the function is defined, while the range is the set of all values of $f(x)$ 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 the domain will be all real numbers except $x=2$ . You can also use the concept of intervals to determine the domain.

Q: What is the difference between a continuous and discontinuous function?

A: A continuous function is a function that can be drawn without lifting the pencil from the paper, while a discontinuous function is a function that has a gap or a break in it. The domain of a continuous function is the set of all real numbers, while the domain of a discontinuous function may be a subset of the real numbers.

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

A: To determine the range of a function, you need to identify the set of all possible output values. For example, if a function is a linear function, then the range will be all real numbers. If a function is a quadratic function, then the range will be all real numbers except the vertex of the parabola.

Q: What is the difference between the maximum and minimum values of a function?

A: The maximum value of a function is the largest possible output value, while the minimum value of a function is the smallest possible output value. The range of a function is the set of all possible output values, including the maximum and minimum values.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on the coordinate plane that satisfy the equation of the function. You can use a graphing calculator or a computer program to graph a 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 us understand the behavior of the function. The domain tells us the set of all possible input values, while the range tells us the set of all possible output values. This information is useful in many real-world applications, such as physics, engineering, and economics.

Conclusion

In conclusion, the domain and range of a function are important concepts in mathematics. The domain is the set of all possible input values, while the range is the set of all possible output values. By understanding the domain and range of a function, we can better understand the behavior of the function and make predictions about its behavior.

Frequently Asked Questions

Q: What is the domain of the function $f(x)=\frac{1}{x-2}$ ? A: The domain of the function is all real numbers except $x=2$ .
Q: What is the range of the function $f(x)=x^2$ ? A: The range of the function is all real numbers except the vertex of the parabola.
Q: How do I determine the domain of a function with a denominator of $x-2$ ? A: To determine the domain of a function with a denominator of $x-2$ , you need to identify the value of $x$ that makes the denominator equal to zero. In this case, the value of $x$ is $x=2$ .
Q: What is the significance of the domain and range of a function in real-world applications? A: The domain and range of a function are important in many real-world applications, such as physics, engineering, and economics. They help us understand the behavior of the function and make predictions about its behavior.

Answer Key

Q: What is the difference between the domain and range of a function? A: The domain is the set of all possible input values, while the range is the set of all possible output values.
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.
Q: What is the difference between a continuous and discontinuous function? A: A continuous function is a function that can be drawn without lifting the pencil from the paper, while a discontinuous function is a function that has a gap or a break in it.
Q: How do I determine the range of a function? A: To determine the range of a function, you need to identify the set of all possible output values.
Q: What is the difference between the maximum and minimum values of a function? A: The maximum value of a function is the largest possible output value, while the minimum value of a function is the smallest possible output value.
Q: How do I graph a function? A: To graph a function, you need to plot the points on the coordinate plane that satisfy the equation of 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 us understand the behavior of the function.