Question
Graph the function \( f(x) = \sqrt[3]{x - 4} \) and identify its domain and range.
Ask by Ortega Adkins. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The graph of \( f(x) = \sqrt[3]{x - 4} \) is a shifted cube root curve passing through (4,0), increasing from left to right. The domain and range are both all real numbers.
Solution
Certainly! Let's analyze and graph the function \( f(x) = \sqrt[3]{x - 4} \), and identify its domain and range.
### Function Analysis
1. **Basic Cube Root Function**:
- The standard cube root function is \( f(x) = \sqrt[3]{x} \).
- Its graph passes through the origin (0,0), with symmetry about the origin.
- It increases continuously from left to right, extending to both positive and negative infinity.
2. **Transformation Applied**:
- The function \( f(x) = \sqrt[3]{x - 4} \) involves a horizontal shift.
- Subtracting 4 inside the cube root shifts the graph **4 units to the right**.
- So, instead of passing through (0,0), it passes through (4,0).
### Graph of \( f(x) = \sqrt[3]{x - 4} \)
Below is a sketch of the graph:
```
y
|
| /
| /
| /
| /
|------------(4,0)-----------
| /
| /
| /
| /
| /
| /
|
+---------------------------- x
```
**Key Features**:
- **Point of Shift**: The graph passes through the point (4,0).
- **Behavior**:
- As \( x \) approaches infinity, \( f(x) \) also approaches infinity.
- As \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.
- The graph is smooth and has no breaks or sharp corners.
### Domain and Range
- **Domain**:
- The domain of a cube root function is all real numbers because you can take the cube root of any real number (positive, negative, or zero).
- **Domain**: \( (-\infty, \infty) \)
- **Range**:
- Similarly, the range is all real numbers since the output of the cube root function can be any real number.
- **Range**: \( (-\infty, \infty) \)
### Summary
- **Graph**: A shifted cube root curve passing through (4,0), increasing from left to right without any restrictions.
- **Domain**: All real numbers \( \mathbb{R} \) or \( (-\infty, \infty) \).
- **Range**: All real numbers \( \mathbb{R} \) or \( (-\infty, \infty) \).
If you have access to graphing tools like Desmos, GeoGebra, or a graphing calculator, you can plot the function \( f(x) = \sqrt[3]{x - 4} \) to visualize its shape more accurately.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To graph the function \( f(x) = \sqrt[3]{x - 4} \), start by plotting key points. When \( x = 4 \), \( f(4) = 0 \). For other values, you can choose \( x = 5 \) (giving \( f(5) = 1 \)) and \( x = 3 \) (resulting in \( f(3) = -1 \)). The function is a cube root transformation, which means it will have a characteristic S-shape, crossing the origin of the adjusted coordinate system. The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \), since you can input any \( x \) value. The range is also all real numbers, \( (-\infty, \infty) \), as cube roots yield real numbers for all inputs. So, feel free to explore left and right, up and down on that graph!
