Comparing and Contrasting Domain and Range

Oct 13, 2025 by windailys.com 110 views

Introduction

When comparing and contrasting two functions, it's essential to consider their domain and 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 two functions, f(x) = 5x and g(x) = 5^x, and discuss which statements could be included in an explanation.

Domain of Functions

Domain of f(x) = 5x

The domain of f(x) = 5x is all real numbers, denoted as (-∞, ∞). This is because for any real number x, we can multiply it by 5, and the result will always be a real number.

Domain of g(x) = 5^x

The domain of g(x) = 5^x is also all real numbers, denoted as (-∞, ∞). This is because for any real number x, we can raise 5 to the power of x, and the result will always be a positive real number.

Range of Functions

Range of f(x) = 5x

The range of f(x) = 5x is all real numbers, denoted as (-∞, ∞). This is because for any real number x, we can multiply it by 5, and the result will always be a real number.

Range of g(x) = 5^x

The range of g(x) = 5^x is all positive real numbers, denoted as (0, ∞). This is because for any real number x, we can raise 5 to the power of x, and the result will always be a positive real number.

Comparing and Contrasting Domain and Range

When comparing and contrasting the domain and range of f(x) = 5x and g(x) = 5^x, we can see that both functions have the same domain, which is all real numbers. However, the range of f(x) = 5x is all real numbers, while the range of g(x) = 5^x is all positive real numbers.

Conclusion

In conclusion, when comparing and contrasting the domain and range of f(x) = 5x and g(x) = 5^x, we can see that both functions have the same domain, but the range of f(x) = 5x is all real numbers, while the range of g(x) = 5^x is all positive real numbers.

Possible Statements

Based on the above discussion, the following statements could be included in an explanation:

The domain of f(x) = 5x and g(x) = 5^x is all real numbers.
The range of f(x) = 5x is all real numbers, while the range of g(x) = 5^x is all positive real numbers.
The domain and range of f(x) = 5x and g(x) = 5^x are different.
The domain of f(x) = 5x and g(x) = 5^x is the same, but the range is different.

Therefore, the two possible statements that could be included in an explanation are:

The domain of f(x) = 5x and g(x) = 5^x is all real numbers.
The range of f(x) = 5x is all real numbers, while the range of g(x) = 5^x is all positive real numbers.
Q&A: Domain and Range of Functions =====================================

Introduction

In our previous article, we discussed the domain and range of two functions, f(x) = 5x and g(x) = 5^x. 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 real numbers for which the function is defined.

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 real numbers that the function can produce.

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 a function has a denominator of x, then x cannot be equal to 0, because division by zero is undefined.

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 y that the function can produce. For example, if a function is a linear function, then its range is all real numbers.

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 of a function is the set of all possible output values. In other words, the domain is the set of all x-values, while the range is the set of all y-values.

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 a function is a linear function, then its domain and range are both all real numbers.

Q: Can the domain of a function be a subset of the real numbers?

A: Yes, the domain of a function can be a subset of the real numbers. For example, if a function is defined only for positive real numbers, then its domain is the set of all positive real numbers.

Q: Can the range of a function be a subset of the real numbers?

A: Yes, the range of a function can be a subset of the real numbers. For example, if a function is a quadratic function, then its range is a subset of the real numbers.

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 y that the function can produce. You can use a graphing calculator or a computer program to graph the function.

Q: Can I have a function with an empty domain?

A: Yes, it is possible to have a function with an empty domain. For example, if a function is defined only for a single value of x, then its domain is empty.

Q: Can I have a function with an empty range?

A: Yes, it is possible to have a function with an empty range. For example, if a function is a constant function, then its range is a single value, and its range is empty.

Conclusion

In conclusion, the domain and range of a function are two 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, you can better understand the behavior of the function and how it relates to the real world.