Answer :
To determine the domain and range of the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex], let's analyze the characteristics of the function step-by-step.
### Step 1: Understanding the Cube Root Function
The cube root function, [tex]\( \sqrt[3]{u} \)[/tex], where [tex]\( u \)[/tex] is any real number, is defined for all real numbers. This means that there are no restrictions on the value of [tex]\( u \)[/tex].
### Step 2: Determine the Domain
The expression inside the cube root is [tex]\( -3x - 9 \)[/tex]. For a cube root function, this expression can take any real number value because the cube root is defined for all real numbers.
Thus, there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex]. This implies that the domain of [tex]\( f(x) \)[/tex] is all real numbers.
In interval notation, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Step 3: Determine the Range
Next, we analyze the range of the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex].
Since the cube root of any real number is also a real number, [tex]\( \sqrt[3]{-3x - 9} \)[/tex] can take any real number as its output. There is no restriction on the output values of a cube root function.
Therefore, the range of [tex]\( f(x) \)[/tex] is all real numbers.
In interval notation, the range is:
[tex]\[ (-\infty, \infty) \][/tex]
### Final Answer:
- Domain: [tex]\((-∞, ∞)\)[/tex]
- Range: [tex]\((-∞, ∞)\)[/tex]
This completes our detailed analysis of the domain and range of the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex].
### Step 1: Understanding the Cube Root Function
The cube root function, [tex]\( \sqrt[3]{u} \)[/tex], where [tex]\( u \)[/tex] is any real number, is defined for all real numbers. This means that there are no restrictions on the value of [tex]\( u \)[/tex].
### Step 2: Determine the Domain
The expression inside the cube root is [tex]\( -3x - 9 \)[/tex]. For a cube root function, this expression can take any real number value because the cube root is defined for all real numbers.
Thus, there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex]. This implies that the domain of [tex]\( f(x) \)[/tex] is all real numbers.
In interval notation, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Step 3: Determine the Range
Next, we analyze the range of the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex].
Since the cube root of any real number is also a real number, [tex]\( \sqrt[3]{-3x - 9} \)[/tex] can take any real number as its output. There is no restriction on the output values of a cube root function.
Therefore, the range of [tex]\( f(x) \)[/tex] is all real numbers.
In interval notation, the range is:
[tex]\[ (-\infty, \infty) \][/tex]
### Final Answer:
- Domain: [tex]\((-∞, ∞)\)[/tex]
- Range: [tex]\((-∞, ∞)\)[/tex]
This completes our detailed analysis of the domain and range of the function [tex]\( f(x) = \sqrt[3]{-3x - 9} \)[/tex].
