Answer :
To address the question related to the function [tex]\( y = \sqrt{3x - 9} \)[/tex], let's break it down into the two parts specified: finding the domain and the range of the function.
### Part (a): Finding the Domain
The domain of a function is the set of all possible input values (in this case, values of [tex]\( x \)[/tex]) that the function can accept without causing any mathematical errors, like division by zero or taking the square root of a negative number.
For the function [tex]\( y = \sqrt{3x - 9} \)[/tex]:
- The expression inside the square root, [tex]\( 3x - 9 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.
#### Step-by-Step:
1. Set up the inequality to ensure the expression inside the square root is non-negative:
[tex]\[ 3x - 9 \geq 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x \geq 9 \][/tex]
[tex]\[ x \geq 3 \][/tex]
Therefore, the domain of the function [tex]\( y = \sqrt{3x - 9} \)[/tex] is:
[tex]\[ x \geq 3 \][/tex]
This means that the function is defined for all [tex]\( x \)[/tex] values starting from 3 and extending to infinity. In interval notation, this is written as:
[tex]\[ [3, \infty) \][/tex]
### Part (b): Finding the Range
The range of a function is the set of all possible output values (in this case values of [tex]\( y \)[/tex]) that the function can produce.
For the function [tex]\( y = \sqrt{3x - 9} \)[/tex]:
- We need to consider how [tex]\( y \)[/tex] behaves as [tex]\( x \)[/tex] varies within its domain.
#### Step-by-Step:
1. Since [tex]\( x \)[/tex] starts at 3 (the minimum value in the domain), substitute [tex]\( x = 3 \)[/tex] to find the minimum value of [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{3(3) - 9} = \sqrt{9 - 9} = \sqrt{0} = 0 \][/tex]
2. As [tex]\( x \)[/tex] increases beyond 3, the term [tex]\( 3x - 9 \)[/tex] becomes positive and grows larger. The square root of a larger and larger positive number also results in larger values of [tex]\( y \)[/tex]. Thus, as [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] also approaches infinity.
Therefore, the range of the function [tex]\( y = \sqrt{3x - 9} \)[/tex] is:
[tex]\[ y \geq 0 \][/tex]
This means that the function can produce output values starting from 0 and extending to infinity. In interval notation, this is written as:
[tex]\[ [0, \infty) \][/tex]
### Summary
#### (a) Domain:
[tex]\[ x \geq 3 \quad \text{or} \quad [3, \infty) \][/tex]
#### (b) Range:
[tex]\[ y \geq 0 \quad \text{or} \quad [0, \infty) \][/tex]
### Part (a): Finding the Domain
The domain of a function is the set of all possible input values (in this case, values of [tex]\( x \)[/tex]) that the function can accept without causing any mathematical errors, like division by zero or taking the square root of a negative number.
For the function [tex]\( y = \sqrt{3x - 9} \)[/tex]:
- The expression inside the square root, [tex]\( 3x - 9 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.
#### Step-by-Step:
1. Set up the inequality to ensure the expression inside the square root is non-negative:
[tex]\[ 3x - 9 \geq 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x \geq 9 \][/tex]
[tex]\[ x \geq 3 \][/tex]
Therefore, the domain of the function [tex]\( y = \sqrt{3x - 9} \)[/tex] is:
[tex]\[ x \geq 3 \][/tex]
This means that the function is defined for all [tex]\( x \)[/tex] values starting from 3 and extending to infinity. In interval notation, this is written as:
[tex]\[ [3, \infty) \][/tex]
### Part (b): Finding the Range
The range of a function is the set of all possible output values (in this case values of [tex]\( y \)[/tex]) that the function can produce.
For the function [tex]\( y = \sqrt{3x - 9} \)[/tex]:
- We need to consider how [tex]\( y \)[/tex] behaves as [tex]\( x \)[/tex] varies within its domain.
#### Step-by-Step:
1. Since [tex]\( x \)[/tex] starts at 3 (the minimum value in the domain), substitute [tex]\( x = 3 \)[/tex] to find the minimum value of [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{3(3) - 9} = \sqrt{9 - 9} = \sqrt{0} = 0 \][/tex]
2. As [tex]\( x \)[/tex] increases beyond 3, the term [tex]\( 3x - 9 \)[/tex] becomes positive and grows larger. The square root of a larger and larger positive number also results in larger values of [tex]\( y \)[/tex]. Thus, as [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] also approaches infinity.
Therefore, the range of the function [tex]\( y = \sqrt{3x - 9} \)[/tex] is:
[tex]\[ y \geq 0 \][/tex]
This means that the function can produce output values starting from 0 and extending to infinity. In interval notation, this is written as:
[tex]\[ [0, \infty) \][/tex]
### Summary
#### (a) Domain:
[tex]\[ x \geq 3 \quad \text{or} \quad [3, \infty) \][/tex]
#### (b) Range:
[tex]\[ y \geq 0 \quad \text{or} \quad [0, \infty) \][/tex]
