For The Following Exercise, Find The Domain Of The Function Using Interval Notation.${ F(x) = \sqrt[3]{5 - 4x} }$

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Domain of a Function: Understanding the Concept

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems or undefined results. In this article, we will explore the concept of domain and how to find the domain of a function using interval notation.

What is Interval Notation?

Interval notation is a way of writing the domain of a function using a specific notation. It is a shorthand way of writing the domain, making it easier to read and understand. Interval notation consists of a combination of numbers and symbols, such as parentheses, brackets, and commas. The most common symbols used in interval notation are:

  • Parentheses: ( ) - used to indicate that the endpoint is not included.
  • Brackets: [ ] - used to indicate that the endpoint is included.
  • Commas: , - used to separate different intervals.

Finding the Domain of a Function

To find the domain of a function, we need to identify the values of x that make the function undefined or produce a non-real result. In the case of the given function, f(x)=5−4x3{ f(x) = \sqrt[3]{5 - 4x} }, we need to find the values of x that make the expression inside the cube root undefined or produce a non-real result.

Step 1: Identify the Values that Make the Expression Undefined

The expression inside the cube root is 5−4x{ 5 - 4x }. This expression is undefined when the value inside the cube root is equal to zero. Therefore, we need to find the value of x that makes 5−4x=0{ 5 - 4x = 0 }.

Step 2: Solve the Equation

To solve the equation 5−4x=0{ 5 - 4x = 0 }, we need to isolate the variable x. We can do this by adding 4x to both sides of the equation and then dividing both sides by 4.

5−4x=0{ 5 - 4x = 0 } 4x=5{ 4x = 5 } x=54{ x = \frac{5}{4} }

Step 3: Determine the Domain

Now that we have found the value of x that makes the expression inside the cube root undefined, we can determine the domain of the function. Since the expression inside the cube root is undefined only when x is equal to 54{ \frac{5}{4} }, the domain of the function is all real numbers except 54{ \frac{5}{4} }.

Writing the Domain in Interval Notation

To write the domain in interval notation, we need to use the symbols and notation we discussed earlier. Since the domain is all real numbers except 54{ \frac{5}{4} }, we can write it as:

(−∞,54)∪(54,∞){ (-\infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty) }

This notation indicates that the domain is all real numbers less than 54{ \frac{5}{4} } and all real numbers greater than 54{ \frac{5}{4} }.

Conclusion

In conclusion, finding the domain of a function using interval notation requires identifying the values of x that make the function undefined or produce a non-real result. We can do this by solving equations and using the symbols and notation of interval notation. By following these steps, we can determine the domain of a function and write it in interval notation.

Example Problems

  • Find the domain of the function f(x)=2x−13{ f(x) = \sqrt[3]{2x - 1} }.
  • Find the domain of the function f(x)=x2−43{ f(x) = \sqrt[3]{x^2 - 4} }.

Solutions

  • The domain of the function f(x)=2x−13{ f(x) = \sqrt[3]{2x - 1} } is all real numbers except 12{ \frac{1}{2} }.
  • The domain of the function f(x)=x2−43{ f(x) = \sqrt[3]{x^2 - 4} } is all real numbers except ±2{ \pm 2 }.

Final Thoughts

Finding the domain of a function using interval notation is an important concept in mathematics. It requires identifying the values of x that make the function undefined or produce a non-real result and writing the domain in interval notation. By following the steps outlined in this article, we can determine the domain of a function and write it in interval notation.
Domain of a Function: Q&A

In the previous article, we discussed the concept of domain and how to find the domain of a function using interval notation. In this article, we will answer some frequently asked questions about the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems or undefined results.

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

A: To find the domain of a function, you need to identify the values of x that make the function undefined or produce a non-real result. You can do this by solving equations and using the symbols and notation of interval notation.

Q: What are the symbols used in interval notation?

A: The most common symbols used in interval notation are:

  • Parentheses: ( ) - used to indicate that the endpoint is not included.
  • Brackets: [ ] - used to indicate that the endpoint is included.
  • Commas: , - used to separate different intervals.

Q: How do I write the domain in interval notation?

A: To write the domain in interval notation, you need to use the symbols and notation we discussed earlier. For example, if the domain is all real numbers except 54{ \frac{5}{4} }, you can write it as:

(−∞,54)∪(54,∞){ (-\infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty) }

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

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-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, you need to identify the values of y that the function can produce. You can do this by analyzing the function and using the symbols and notation of interval notation.

Q: What are some common mistakes to avoid when finding the domain of a function?

A: Some common mistakes to avoid when finding the domain of a function include:

  • Not considering all possible values of x.
  • Not using the correct symbols and notation.
  • Not solving equations correctly.

Q: Can you give me some examples of functions with different domains?

A: Yes, here are some examples of functions with different domains:

  • f(x)=x3{ f(x) = \sqrt[3]{x} } - the domain is all real numbers.
  • f(x)=1x{ f(x) = \frac{1}{x} } - the domain is all real numbers except 0.
  • f(x)=x{ f(x) = \sqrt{x} } - the domain is all non-negative real numbers.

Q: How do I determine if a function is defined at a particular point?

A: To determine if a function is defined at a particular point, you need to check if the function is continuous at that point. If the function is continuous at that point, then it is defined at that point.

Q: Can you give me some examples of functions that are not defined at a particular point?

A: Yes, here are some examples of functions that are not defined at a particular point:

  • f(x)=1x{ f(x) = \frac{1}{x} } - not defined at x = 0.
  • f(x)=x{ f(x) = \sqrt{x} } - not defined at x < 0.
  • f(x)=log⁡(x){ f(x) = \log(x) } - not defined at x ≤ 0.

Conclusion

In conclusion, finding the domain of a function using interval notation is an important concept in mathematics. By understanding the symbols and notation used in interval notation and following the steps outlined in this article, you can determine the domain of a function and write it in interval notation.