Answer :
Let’s analyze and understand the function [tex]\( f(x) = -\sqrt[3]{x-3} - 1 \)[/tex] and determine its behavior to match it with a specific graph.
### Step 1: Transformations
The function [tex]\( f(x) \)[/tex] can be rewritten with its transformations clearly separated:
[tex]\[ f(x) = -\sqrt[3]{x-3} - 1 \][/tex]
#### Key Transformations:
1. [tex]\( x - 3 \)[/tex]:
- This represents a horizontal shift 3 units to the right.
2. [tex]\( \sqrt[3]{x-3} \)[/tex]:
- This represents a cubic root, which affects the shape of the graph (makes it more S-shaped).
3. [tex]\( -\sqrt[3]{x-3} \)[/tex]:
- This indicates a reflection over the x-axis.
4. [tex]\( -\sqrt[3]{x-3} - 1 \)[/tex]:
- This is a vertical shift 1 unit down.
### Step 2: Critical Points and Behavior
1. Y-intercept:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sqrt[3]{0-3} - 1 = -\sqrt[3]{-3} - 1 \][/tex]
- The cube root of [tex]\(-3\)[/tex] is a negative real number.
[tex]\[ \sqrt[3]{-3} \approx -1.442 \][/tex]
Thus,
[tex]\[ f(0) \approx -(-1.442) - 1 \approx 1.442 - 1 = 0.442 \][/tex]
2. Asymptotic Behavior:
- The cube root function is defined for all real numbers, so there are no vertical or horizontal asymptotes.
3. Other Values:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -\sqrt[3]{3-3} - 1 = -\sqrt[3]{0} - 1 = -0 - 1 = -1 \][/tex]
- For other values from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] (as per the result given):
[tex]\[ \begin{align*} f(-10) & \approx -(-2.1757 + 2.0363i) - 1 \\ f(-9) & \approx -(-2.1447 + 1.9827i) - 1 \\ f(-8) & \approx -(-2.1120 + 1.9260i) - 1 \\ f(-7) & \approx -(-2.0772 + 1.8658i) - 1 \\ f(-6) & \approx -(-2.0400 + 1.8014i) - 1 \\ f(-5) & = -2 - 1 \\ f(-4) & \approx -(-1.9565 + 1.6566i) - 1 \\ f(-3) & \approx -(-1.9086 + 1.5737i) - 1 \\ f(-2) & \approx -(-1.8550 + 1.4809i) - 1 \\ f(-1) & \approx -(-1.7937 + 1.3747i) - 1 \\ f(0) & \approx -(-1.7211 + 1.2490i) - 1 \\ f(1) & \approx -(-1.6300 + 1.0911i) - 1 \\ f(2) & \approx -(-1.5) - 1 \approx 1 - 1 = 0 \\ f(3) & = -1 - 1 = -2 \\ f(4) & \approx -(-2.2599) - 1 \approx 3.2599 - 1 \approx 2.2600 \\ f(5) & \approx -(-2.4422) - 1 \approx 3.4422 - 1 \approx 2.4422 \\ f(6) & \approx -(-2.5874) - 1 \approx 3.5874 - 1 \approx 2.5874 \\ f(7) & \approx -(-2.7100) - 1 \approx 3.7100 - 1 \approx 2.7100 \\ f(8) & \approx -(-2.8171) - 1 \approx 3.8171 - 1 \approx 2.8171 \\ f(9) & \approx -(-2.9129) - 1 \approx 3.9129 - 1 \approx 2.9129 \\ f(10) & \approx -(-3) - 1 \approx 4 - 1 = 3 \end{align*} \][/tex]
- The graph will be a transformed version of the [tex]\( \sqrt[3]{x} \)[/tex] curve, shifted right by 3 units, reflected about the x-axis, and shifted down by 1 unit.
### Step 3: Conclusion and Graph Matching
- The points and transformations suggest that the graph behaves with an S-shape shifted rightwards and downwards. It transitions smoothly from positive to negative values with the specified shifts and reflections.
By matching this description with the available graph options, we select the one representing these transformations. This graph should have a vertical asymptotic behavior fitting the given transformations and plotting behavior.
### Step 1: Transformations
The function [tex]\( f(x) \)[/tex] can be rewritten with its transformations clearly separated:
[tex]\[ f(x) = -\sqrt[3]{x-3} - 1 \][/tex]
#### Key Transformations:
1. [tex]\( x - 3 \)[/tex]:
- This represents a horizontal shift 3 units to the right.
2. [tex]\( \sqrt[3]{x-3} \)[/tex]:
- This represents a cubic root, which affects the shape of the graph (makes it more S-shaped).
3. [tex]\( -\sqrt[3]{x-3} \)[/tex]:
- This indicates a reflection over the x-axis.
4. [tex]\( -\sqrt[3]{x-3} - 1 \)[/tex]:
- This is a vertical shift 1 unit down.
### Step 2: Critical Points and Behavior
1. Y-intercept:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sqrt[3]{0-3} - 1 = -\sqrt[3]{-3} - 1 \][/tex]
- The cube root of [tex]\(-3\)[/tex] is a negative real number.
[tex]\[ \sqrt[3]{-3} \approx -1.442 \][/tex]
Thus,
[tex]\[ f(0) \approx -(-1.442) - 1 \approx 1.442 - 1 = 0.442 \][/tex]
2. Asymptotic Behavior:
- The cube root function is defined for all real numbers, so there are no vertical or horizontal asymptotes.
3. Other Values:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -\sqrt[3]{3-3} - 1 = -\sqrt[3]{0} - 1 = -0 - 1 = -1 \][/tex]
- For other values from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] (as per the result given):
[tex]\[ \begin{align*} f(-10) & \approx -(-2.1757 + 2.0363i) - 1 \\ f(-9) & \approx -(-2.1447 + 1.9827i) - 1 \\ f(-8) & \approx -(-2.1120 + 1.9260i) - 1 \\ f(-7) & \approx -(-2.0772 + 1.8658i) - 1 \\ f(-6) & \approx -(-2.0400 + 1.8014i) - 1 \\ f(-5) & = -2 - 1 \\ f(-4) & \approx -(-1.9565 + 1.6566i) - 1 \\ f(-3) & \approx -(-1.9086 + 1.5737i) - 1 \\ f(-2) & \approx -(-1.8550 + 1.4809i) - 1 \\ f(-1) & \approx -(-1.7937 + 1.3747i) - 1 \\ f(0) & \approx -(-1.7211 + 1.2490i) - 1 \\ f(1) & \approx -(-1.6300 + 1.0911i) - 1 \\ f(2) & \approx -(-1.5) - 1 \approx 1 - 1 = 0 \\ f(3) & = -1 - 1 = -2 \\ f(4) & \approx -(-2.2599) - 1 \approx 3.2599 - 1 \approx 2.2600 \\ f(5) & \approx -(-2.4422) - 1 \approx 3.4422 - 1 \approx 2.4422 \\ f(6) & \approx -(-2.5874) - 1 \approx 3.5874 - 1 \approx 2.5874 \\ f(7) & \approx -(-2.7100) - 1 \approx 3.7100 - 1 \approx 2.7100 \\ f(8) & \approx -(-2.8171) - 1 \approx 3.8171 - 1 \approx 2.8171 \\ f(9) & \approx -(-2.9129) - 1 \approx 3.9129 - 1 \approx 2.9129 \\ f(10) & \approx -(-3) - 1 \approx 4 - 1 = 3 \end{align*} \][/tex]
- The graph will be a transformed version of the [tex]\( \sqrt[3]{x} \)[/tex] curve, shifted right by 3 units, reflected about the x-axis, and shifted down by 1 unit.
### Step 3: Conclusion and Graph Matching
- The points and transformations suggest that the graph behaves with an S-shape shifted rightwards and downwards. It transitions smoothly from positive to negative values with the specified shifts and reflections.
By matching this description with the available graph options, we select the one representing these transformations. This graph should have a vertical asymptotic behavior fitting the given transformations and plotting behavior.
