What Are The Domain And Range Of The Function?$f(x)=3 \sqrt{x-3}$A. Domain: $(3, \infty$\] Range: $(3, \infty$\]B. Domain: \[8, \infty$\] Range: $(0, \infty$\]C. Domain: \[3, \infty$\] Range:

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Introduction

In mathematics, functions are used to describe the relationship between variables. The domain and range of a function are two essential concepts that help us understand the behavior of the function. In this article, we will explore the domain and range of the function f(x)=3xβˆ’3f(x) = 3\sqrt{x-3} and provide a step-by-step guide on how to determine the domain and range of a function.

What are Domain and Range?

The domain of a function is the set of all possible input values (x-values) that the function can accept. It is the set of all possible values of x for which the function is defined. On the other hand, the range of a function is the set of all possible output values (y-values) that the function can produce.

Determining the Domain of a Function

To determine the domain of a function, we need to consider the values of x that make the function undefined. In the case of the function f(x)=3xβˆ’3f(x) = 3\sqrt{x-3}, the function is undefined when the expression inside the square root is negative, i.e., when xβˆ’3<0x-3 < 0. This is because the square root of a negative number is not a real number.

Step 1: Find the Value of x that Makes the Expression Inside the Square Root Negative

To find the value of x that makes the expression inside the square root negative, we need to solve the inequality xβˆ’3<0x-3 < 0. This can be done by adding 3 to both sides of the inequality, which gives us x<3x < 3.

Step 2: Determine the Domain of the Function

Since the function is undefined when x<3x < 3, the domain of the function is all real numbers greater than 3. This can be represented as (3,∞)(3, \infty).

Determining the Range of a Function

To determine the range of a function, we need to consider the values of y that the function can produce. In the case of the function f(x)=3xβˆ’3f(x) = 3\sqrt{x-3}, the function is always positive because the square root of a number is always non-negative, and multiplying it by 3 makes it positive.

Step 1: Determine the Minimum Value of y

Since the function is always positive, the minimum value of y is 0. This is because the square root of 0 is 0, and multiplying it by 3 gives us 0.

Step 2: Determine the Maximum Value of y

Since the function is always increasing as x increases, the maximum value of y is unbounded. This means that y can take on any positive value.

Step 3: Determine the Range of the Function

Since the minimum value of y is 0 and the maximum value of y is unbounded, the range of the function is all real numbers greater than 0. This can be represented as (0,∞)(0, \infty).

Conclusion

In conclusion, the domain of the function f(x)=3xβˆ’3f(x) = 3\sqrt{x-3} is (3,∞)(3, \infty), and the range of the function is (0,∞)(0, \infty). By following the steps outlined in this article, you can determine the domain and range of any function.

Common Mistakes to Avoid

When determining the domain and range of a function, there are several common mistakes to avoid:

  • Not considering the values of x that make the function undefined: Make sure to consider the values of x that make the function undefined, such as when the expression inside the square root is negative.
  • Not considering the values of y that the function can produce: Make sure to consider the values of y that the function can produce, such as when the function is always positive.
  • Not using interval notation correctly: Make sure to use interval notation correctly when representing the domain and range of a function.

Practice Problems

To practice determining the domain and range of a function, try the following problems:

  • Determine the domain and range of the function f(x)=2x2+1f(x) = 2x^2 + 1.
  • Determine the domain and range of the function f(x)=xβˆ’2f(x) = \sqrt{x-2}.
  • Determine the domain and range of the function f(x)=x3βˆ’2f(x) = x^3 - 2.

Answer Key

  • The domain of the function f(x)=2x2+1f(x) = 2x^2 + 1 is (βˆ’βˆž,∞)(-\infty, \infty), and the range of the function is [1,∞)[1, \infty).
  • The domain of the function f(x)=xβˆ’2f(x) = \sqrt{x-2} is [2,∞)[2, \infty), and the range of the function is [0,∞)[0, \infty).
  • The domain of the function f(x)=x3βˆ’2f(x) = x^3 - 2 is (βˆ’βˆž,∞)(-\infty, \infty), and the range of the function is (βˆ’βˆž,∞)(-\infty, \infty).

Conclusion

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept. It is the set of all possible values of x 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 (y-values) 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 that make the function undefined. This can include values that make the denominator of a fraction equal to zero, or values that make the expression inside a square root negative.

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. This can include values that are always positive, or values that are always negative.

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

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. This is represented as (βˆ’βˆž,∞)(-\infty, \infty).

Q: Can a function have a range of all real numbers?

A: Yes, a function can have a range of all real numbers. This is represented as (βˆ’βˆž,∞)(-\infty, \infty).

Q: How do I represent the domain and range of a function using interval notation?

A: To represent the domain and range of a function using interval notation, you need to use the following notation:

  • (a,b)(a, b) represents an open interval, where aa and bb are the endpoints of the interval.
  • [a,b][a, b] represents a closed interval, where aa and bb are the endpoints of the interval.
  • (βˆ’βˆž,a)(-\infty, a) represents an interval that extends to negative infinity and includes the value aa.
  • (a,∞)(a, \infty) represents an interval that extends to positive infinity and includes the value aa.

Q: What is the difference between an open and closed interval?

A: An open interval is an interval that does not include the endpoints, while a closed interval is an interval that includes the endpoints.

Q: Can a function have a domain and range that are the same?

A: Yes, a function can have a domain and range that are the same. This is represented as (βˆ’βˆž,∞)(-\infty, \infty).

Q: How do I determine the domain and range of a function with a square root?

A: To determine the domain and range of a function with a square root, you need to consider the values of x that make the expression inside the square root negative. This can be done by solving the inequality xβˆ’3<0x-3 < 0.

Q: How do I determine the domain and range of a function with a fraction?

A: To determine the domain and range of a function with a fraction, you need to consider the values of x that make the denominator of the fraction equal to zero. This can be done by solving the equation xβˆ’3=0x-3 = 0.

Q: Can a function have a domain and range that are not the same?

A: Yes, a function can have a domain and range that are not the same. For example, the function f(x)=xβˆ’3f(x) = \sqrt{x-3} has a domain of [3,∞)[3, \infty) and a range of [0,∞)[0, \infty).

Conclusion

In conclusion, the domain and range of a function are essential concepts in mathematics. By understanding how to determine the domain and range of a function, you can better understand the behavior of the function and make predictions about its output. Remember to consider the values of x that make the function undefined and the values of y that the function can produce. With practice, you will become proficient in determining the domain and range of a function.