What Are The Domain And Range Of The Function?$f(x) = \frac{3}{5x^5}$A. Domain: $(-\infty, 0) \cup (0, \infty$\] Range: $(0, \infty$\]B. Domain: $(-\infty, \infty$\] Range: $(-\infty, 0$\]C. Domain:

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**Understanding the Domain and Range of a Function: A Comprehensive Guide**

What are the Domain and Range of the Function?

When dealing with functions, it's essential to understand the concept of domain and range. 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'll explore the domain and range of the function f(x)=35x5f(x) = \frac{3}{5x^5} and provide a comprehensive guide to understanding these concepts.

What is the Domain of the Function?

The domain of a function is the set of all possible input values (x-values) that the function can accept. In the case of the function f(x)=35x5f(x) = \frac{3}{5x^5}, we need to determine the values of x that will not result in division by zero.

To find the domain, we need to identify the values of x that will make the denominator (5x^5) equal to zero. Since the denominator is a polynomial expression, we can set it equal to zero and solve for x:

5x^5 = 0

x^5 = 0

x = 0

Therefore, the value of x that will make the denominator equal to zero is x = 0. However, since the function is undefined at x = 0, we need to exclude this value from the domain.

The domain of the function is the set of all real numbers except x = 0. This can be represented as:

Domain: (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty)

What is the Range of the Function?

The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range, we need to determine the possible values of y that the function can produce.

Since the function is a rational function, we can analyze the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, the function approaches zero. As x approaches negative infinity, the function also approaches zero.

However, since the function is a rational function, we need to consider the behavior of the function as x approaches zero. As x approaches zero from the left (negative values), the function approaches negative infinity. As x approaches zero from the right (positive values), the function approaches positive infinity.

Therefore, the range of the function is the set of all positive real numbers. This can be represented as:

Range: (0,∞)(0, \infty)

Q&A: Domain and Range of the Function

Q: What is the domain of the function f(x)=35x5f(x) = \frac{3}{5x^5}?

A: The domain of the function is the set of all real numbers except x = 0. This can be represented as: Domain: (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty)

Q: What is the range of the function f(x)=35x5f(x) = \frac{3}{5x^5}?

A: The range of the function is the set of all positive real numbers. This can be represented as: Range: (0,∞)(0, \infty)

Q: Why is the function undefined at x = 0?

A: The function is undefined at x = 0 because the denominator (5x^5) is equal to zero at this value. Division by zero is undefined in mathematics.

Q: How can I determine the domain and range of a function?

A: To determine the domain and range of a function, you need to analyze the behavior of the function as x approaches positive and negative infinity, and as x approaches zero. You also need to consider any restrictions on the domain, such as division by zero.

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 is the set of all possible output values (y-values) that the function can produce.

Conclusion

In conclusion, the domain and range of a function are essential concepts in mathematics that help us understand the behavior of a function. By analyzing the behavior of the function as x approaches positive and negative infinity, and as x approaches zero, we can determine the domain and range of the function. In this article, we've explored the domain and range of the function f(x)=35x5f(x) = \frac{3}{5x^5} and provided a comprehensive guide to understanding these concepts.

Frequently Asked Questions

  • What is the domain of the function f(x)=35x5f(x) = \frac{3}{5x^5}? The domain of the function is the set of all real numbers except x = 0. This can be represented as: Domain: (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty)
  • What is the range of the function f(x)=35x5f(x) = \frac{3}{5x^5}? The range of the function is the set of all positive real numbers. This can be represented as: Range: (0,∞)(0, \infty)
  • Why is the function undefined at x = 0? The function is undefined at x = 0 because the denominator (5x^5) is equal to zero at this value. Division by zero is undefined in mathematics.
  • How can I determine the domain and range of a function? To determine the domain and range of a function, you need to analyze the behavior of the function as x approaches positive and negative infinity, and as x approaches zero. You also need to consider any restrictions on the domain, such as division by zero.
  • What is the difference between the domain and range of a 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.