Determine The Domain Of The Function. A. $[-5,1]$ B. $(-\infty, 9]$ C. $ ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) [/tex] D. $(-7,3)$

by ADMIN 143 views

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. It is a crucial concept in understanding the behavior and properties of functions. In this article, we will explore the different types of domains and provide a step-by-step guide on how to determine the domain of a function.

What is the Domain of a Function?

The domain of a function is the set of all possible input values, or x-values, that can be plugged into the function without resulting in an undefined or imaginary output. In other words, it is the set of all possible values of x for which the function is defined.

Types of Domains

There are three main types of domains:

  • Bounded Domain: A bounded domain is a set of values that is limited by a specific range or interval. For example, the domain of the function f(x) = 1/x is (-∞, 0) ∪ (0, ∞), which is a bounded domain.
  • Unbounded Domain: An unbounded domain is a set of values that extends infinitely in one or both directions. For example, the domain of the function f(x) = x^2 is (-∞, ∞), which is an unbounded domain.
  • Discrete Domain: A discrete domain is a set of values that consists of isolated points or intervals. For example, the domain of the function f(x) = x^2 is the set of all integers, which is a discrete domain.

How to Determine the Domain of a Function

To determine the domain of a function, follow these steps:

  1. Check for Division by Zero: Check if the function involves division by zero. If it does, the domain will be restricted to exclude the value that makes the denominator zero.
  2. Check for Square Roots: Check if the function involves square roots. If it does, the domain will be restricted to exclude negative values under the square root.
  3. Check for Logarithms: Check if the function involves logarithms. If it does, the domain will be restricted to exclude negative values inside the logarithm.
  4. Check for Other Restrictions: Check if the function involves other restrictions, such as absolute values or trigonometric functions. If it does, the domain will be restricted accordingly.

Examples of Determining the Domain of a Function

Example 1: f(x) = 1/x

To determine the domain of the function f(x) = 1/x, we need to check for division by zero. The function involves division by x, so we need to exclude the value that makes the denominator zero. In this case, x cannot be equal to zero, so the domain is (-∞, 0) ∪ (0, ∞).

Example 2: f(x) = √(x-2)

To determine the domain of the function f(x) = √(x-2), we need to check for square roots. The function involves a square root, so we need to exclude negative values under the square root. In this case, x-2 cannot be negative, so x must be greater than or equal to 2. The domain is [2, ∞).

Example 3: f(x) = log(x)

To determine the domain of the function f(x) = log(x), we need to check for logarithms. The function involves a logarithm, so we need to exclude negative values inside the logarithm. In this case, x cannot be negative, so the domain is (0, ∞).

Example 4: f(x) = |x-2|

To determine the domain of the function f(x) = |x-2|, we need to check for absolute values. The function involves an absolute value, so the domain is all real numbers.

Conclusion

In conclusion, determining the domain of a function is a crucial step in understanding the behavior and properties of functions. By following the steps outlined in this article, you can determine the domain of a function and identify the type of domain it has. Remember to check for division by zero, square roots, logarithms, and other restrictions to ensure that you have the correct domain.

Common Mistakes to Avoid

  • Not checking for division by zero: Failing to check for division by zero can result in an incorrect domain.
  • Not checking for square roots: Failing to check for square roots can result in an incorrect domain.
  • Not checking for logarithms: Failing to check for logarithms can result in an incorrect domain.
  • Not checking for other restrictions: Failing to check for other restrictions, such as absolute values or trigonometric functions, can result in an incorrect domain.

Final Thoughts

Determining the domain of a function is a critical step in understanding the behavior and properties of functions. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you have the correct domain and can proceed with confidence in your mathematical endeavors.

References

Additional Resources

Frequently Asked Questions

  • Q: What is the domain of a function? A: The domain of a function is the set of all possible input values for which the function is defined.
  • Q: How do I determine the domain of a function? A: To determine the domain of a function, check for division by zero, square roots, logarithms, and other restrictions.
  • Q: What are the different types of domains? A: The different types of domains are bounded, unbounded, and discrete domains.
    Domain of a Function: Frequently Asked Questions =====================================================

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. It is a crucial concept in understanding the behavior and properties of functions.

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

A: To determine the domain of a function, follow these steps:

  1. Check for Division by Zero: Check if the function involves division by zero. If it does, the domain will be restricted to exclude the value that makes the denominator zero.
  2. Check for Square Roots: Check if the function involves square roots. If it does, the domain will be restricted to exclude negative values under the square root.
  3. Check for Logarithms: Check if the function involves logarithms. If it does, the domain will be restricted to exclude negative values inside the logarithm.
  4. Check for Other Restrictions: Check if the function involves other restrictions, such as absolute values or trigonometric functions. If it does, the domain will be restricted accordingly.

Q: What are the different types of domains?

A: The different types of domains are:

  • Bounded Domain: A bounded domain is a set of values that is limited by a specific range or interval. For example, the domain of the function f(x) = 1/x is (-∞, 0) ∪ (0, ∞), which is a bounded domain.
  • Unbounded Domain: An unbounded domain is a set of values that extends infinitely in one or both directions. For example, the domain of the function f(x) = x^2 is (-∞, ∞), which is an unbounded domain.
  • Discrete Domain: A discrete domain is a set of values that consists of isolated points or intervals. For example, the domain of the function f(x) = x^2 is the set of all integers, which is a discrete domain.

Q: How do I find the domain of a function with a square root?

A: To find the domain of a function with a square root, follow these steps:

  1. Check if the value under the square root is negative: If the value under the square root is negative, the domain will be restricted to exclude that value.
  2. Check if the value under the square root is zero: If the value under the square root is zero, the domain will be restricted to exclude that value.
  3. Check if the value under the square root is positive: If the value under the square root is positive, the domain will include all real numbers.

Q: How do I find the domain of a function with a logarithm?

A: To find the domain of a function with a logarithm, follow these steps:

  1. Check if the value inside the logarithm is negative: If the value inside the logarithm is negative, the domain will be restricted to exclude that value.
  2. Check if the value inside the logarithm is zero: If the value inside the logarithm is zero, the domain will be restricted to exclude that value.
  3. Check if the value inside the logarithm is positive: If the value inside the logarithm is positive, the domain will include all real numbers.

Q: How do I find the domain of a function with an absolute value?

A: To find the domain of a function with an absolute value, follow these steps:

  1. Check if the value inside the absolute value is negative: If the value inside the absolute value is negative, the domain will be restricted to exclude that value.
  2. Check if the value inside the absolute value is zero: If the value inside the absolute value is zero, the domain will include all real numbers.
  3. Check if the value inside the absolute value is positive: If the value inside the absolute value is positive, the domain will include all real numbers.

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

A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values for which the function is defined.

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

A: To find the range of a function, follow these steps:

  1. Check if the function is increasing or decreasing: If the function is increasing, the range will include all values greater than or equal to the minimum value. If the function is decreasing, the range will include all values less than or equal to the maximum value.
  2. Check if the function has a maximum or minimum value: If the function has a maximum or minimum value, the range will include all values greater than or equal to the maximum value or less than or equal to the minimum value.
  3. Check if the function has a vertical asymptote: If the function has a vertical asymptote, the range will be restricted to exclude the value that makes the denominator zero.

Conclusion

In conclusion, determining the domain of a function is a crucial step in understanding the behavior and properties of functions. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you have the correct domain and can proceed with confidence in your mathematical endeavors.

Common Mistakes to Avoid

  • Not checking for division by zero: Failing to check for division by zero can result in an incorrect domain.
  • Not checking for square roots: Failing to check for square roots can result in an incorrect domain.
  • Not checking for logarithms: Failing to check for logarithms can result in an incorrect domain.
  • Not checking for other restrictions: Failing to check for other restrictions, such as absolute values or trigonometric functions, can result in an incorrect domain.

Final Thoughts

Determining the domain of a function is a critical step in understanding the behavior and properties of functions. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you have the correct domain and can proceed with confidence in your mathematical endeavors.

References

Additional Resources