Find The Domain Of The Function. F ( T ) = T − 5 3 F(t)=\sqrt[3]{t-5} F ( T ) = 3 T − 5 ​

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable that the function can accept without resulting in an undefined or imaginary output. In this article, we will explore the concept of the domain of a function, focusing on the specific function $f(t)=\sqrt[3]{t-5}$.

What is the Domain of a Function?

The domain of a function is a crucial concept in mathematics, as it determines the range of values that the function can accept. A function is said to be defined for a particular value of the variable if it produces a real number as output. In other words, the function is defined for all values of the variable that do not result in an undefined or imaginary output.

Properties of the Domain

The domain of a function has several important properties:

• Domain is a set: The domain of a function is a set of all possible input values for which the function is defined.
• Domain is not necessarily an interval: The domain of a function can be a set of discrete values, an interval, or a combination of both.
• Domain can be infinite: The domain of a function can be infinite, meaning that it can accept an infinite number of input values.

Finding the Domain of a Function

To find the domain of a function, we need to identify the values of the variable that result in an undefined or imaginary output. In the case of the function $f(t)=\sqrt[3]{t-5}$, we need to find the values of $t$ that result in an undefined or imaginary output.

Step 1: Identify the values that result in an undefined output

The function $f(t)=\sqrt[3]{t-5}$ is defined for all values of $t$ except when $t-5=0$. This is because the cube root of zero is undefined.

Step 2: Solve the equation $t-5=0$

To find the value of $t$ that results in an undefined output, we need to solve the equation $t-5=0$. This equation can be solved by adding 5 to both sides, resulting in $t=5$.

Step 3: Determine the domain of the function

Since the function $f(t)=\sqrt[3]{t-5}$ is defined for all values of $t$ except when $t=5$, the domain of the function is all real numbers except 5.

Conclusion

In conclusion, the domain of a function is the set of all possible input values for which the function is defined. The domain of a function can be a set of discrete values, an interval, or a combination of both. To find the domain of a function, we need to identify the values of the variable that result in an undefined or imaginary output. In the case of the function $f(t)=\sqrt[3]{t-5}$, the domain is all real numbers except 5.

Domain of the Function $f(t)=\sqrt[3]{t-5}$

The domain of the function $f(t)=\sqrt[3]{t-5}$ is all real numbers except 5.

Example Problems

Problem 1

Find the domain of the function $f(x)=\sqrt[3]{x-2}$.

Solution

The function $f(x)=\sqrt[3]{x-2}$ is defined for all values of $x$ except when $x-2=0$. This is because the cube root of zero is undefined. Solving the equation $x-2=0$ results in $x=2$. Therefore, the domain of the function $f(x)=\sqrt[3]{x-2}$ is all real numbers except 2.

Problem 2

Find the domain of the function $f(t)=\sqrt[3]{t+3}$.

Solution

The function $f(t)=\sqrt[3]{t+3}$ is defined for all values of $t$ except when $t+3=0$. This is because the cube root of zero is undefined. Solving the equation $t+3=0$ results in $t=-3$. Therefore, the domain of the function $f(t)=\sqrt[3]{t+3}$ is all real numbers except -3.

Key Takeaways

• The domain of a function is the set of all possible input values for which the function is defined.
• The domain of a function can be a set of discrete values, an interval, or a combination of both.
• To find the domain of a function, we need to identify the values of the variable that result in an undefined or imaginary output.
• The domain of a function can be infinite, meaning that it can accept an infinite number of input values.

Final Thoughts

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. In other words, it is the set of all possible values of the variable that the function can accept without resulting in an undefined or imaginary output.

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 the variable that result in an undefined or imaginary output. This can be done by:

• Identifying any values that make the denominator of a fraction equal to zero.
• Identifying any values that make the square root of a negative number.
• Identifying any values that make the cube root of a negative number.

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, while the range of a function is the set of all possible output values that the function can produce.

Q: Can the domain of a function be infinite?

A: Yes, the domain of a function can be infinite. This means that the function can accept an infinite number of input values.

Q: Can the domain of a function be a set of discrete values?

A: Yes, the domain of a function can be a set of discrete values. This means that the function can only accept a specific set of values, rather than all possible values.

Q: Can the domain of a function be an interval?

A: Yes, the domain of a function can be an interval. This means that the function can accept all values within a specific range.

Q: How do I determine if a function is defined for a particular value?

A: To determine if a function is defined for a particular value, you need to check if the function produces a real number as output. If the function produces a real number, then it is defined for that value.

Q: What is the domain of the function $f(x)=\frac{1}{x-2}$?

A: The domain of the function $f(x)=\frac{1}{x-2}$ is all real numbers except 2. This is because the denominator of the fraction cannot be equal to zero.

Q: What is the domain of the function $f(t)=\sqrt{t+3}$?

A: The domain of the function $f(t)=\sqrt{t+3}$ is all real numbers except -3. This is because the square root of a negative number is undefined.

Q: What is the domain of the function $f(x)=\sqrt[3]{x-2}$?

A: The domain of the function $f(x)=\sqrt[3]{x-2}$ is all real numbers. This is because the cube root of any number is defined.

Q: Can the domain of a function be a combination of discrete values and intervals?

A: Yes, the domain of a function can be a combination of discrete values and intervals. This means that the function can accept a specific set of values, as well as all values within a specific range.

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

A: To graph the domain of a function, you need to identify the values of the variable that result in an undefined or imaginary output. You can then use this information to create a graph that shows the domain of the function.

Q: What is the importance of understanding the domain of a function?

A: Understanding the domain of a function is important because it helps you to:

• Identify the values of the variable that result in an undefined or imaginary output.
• Create a graph that shows the domain of the function.
• Analyze and solve mathematical problems.

Conclusion

In conclusion, the domain of a function is a crucial concept in mathematics that determines the range of values that the function can accept. By understanding the concept of the domain of a function, you can better analyze and solve mathematical problems. The domain of a function can be a set of discrete values, an interval, or a combination of both. To find the domain of a function, you need to identify the values of the variable that result in an undefined or imaginary output.