Answer
The function \( f(x) = \sqrt[3]{x - 4} \) is a cube root function shifted 4 units to the right. Its graph is a smooth curve passing through the point (4,0) and extending infinitely in both directions. The domain and range of the function are all real numbers.
Domain: All real numbers
Range: All real numbers
Solution
Certainly! Let's analyze and graph the function \( f(x) = \sqrt[3]{x - 4} \), and determine its **domain** and **range**.
## Function Overview
The function \( f(x) = \sqrt[3]{x - 4} \) is a **cube root function**. It's a transformation of the basic cube root function \( f(x) = \sqrt[3]{x} \), shifted horizontally.
### Key Characteristics:
- **Base Function**: \( f(x) = \sqrt[3]{x} \)
- **Transformation**: Shifted **4 units to the right**.
## Graphing \( f(x) = \sqrt[3]{x - 4} \)
Here's how you can sketch the graph:
1. **Start with the Base Graph**: The basic cube root graph \( f(x) = \sqrt[3]{x} \) passes through the origin (0,0), with the graph increasing from left to right, flattening as it approaches the x-axis for negative \( x \) and steepening for positive \( x \).
2. **Apply the Horizontal Shift**: Shift the entire graph **4 units to the right** to account for the \( (x - 4) \) inside the cube root.
3. **Plot Key Points**:
- **Original Point**: (0, 0) becomes (4, 0).
- **Additional Points**:
- For \( x = 1 \): \( f(1) = \sqrt[3]{1 - 4} = \sqrt[3]{-3} \approx -1.442 \) → (1, -1.442)
- For \( x = 8 \): \( f(8) = \sqrt[3]{8 - 4} = \sqrt[3]{4} \approx 1.587 \) → (8, 1.587)
4. **Shape of the Graph**:
- The graph passes through all points smoothly, extending infinitely in both directions.
- It has **no asymptotes** and is **continuous** everywhere.
### Visual Representation
While I can't provide a visual graph here, you can visualize the graph as follows:
- **Origin Shift**: Instead of passing through (0,0), it passes through (4,0).
- **Increasing Nature**: As \( x \) increases beyond 4, \( f(x) \) increases; as \( x \) decreases below 4, \( f(x) \) decreases.
- **Symmetry**: The cube root function is symmetric about the origin before shifting. After shifting, this symmetry is around the point (4,0).
## Domain and Range
### Domain
The **domain** of a function is the set of all possible input values (x-values) for which the function is defined.
- **Cube Root Function Properties**: The cube root of any real number is defined (unlike even roots, which require non-negative inputs).
- **For \( f(x) = \sqrt[3]{x - 4} \)**:
- **Domain**: All real numbers.
\[
\boxed{\text{Domain}: \mathbb{R} \text{ (all real numbers)}}
\]
### Range
The **range** of a function is the set of all possible output values (y-values).
- **Cube Root Function Properties**: Since the cube root can produce any real number, so can \( f(x) = \sqrt[3]{x - 4} \).
- **Range**: All real numbers.
\[
\boxed{\text{Range}: \mathbb{R} \text{ (all real numbers)}}
\]
## Summary
- **Function**: \( f(x) = \sqrt[3]{x - 4} \)
- **Graph**: A cube root curve shifted 4 units to the right, passing through (4,0), extending infinitely in all directions.
- **Domain**: All real numbers (\( \mathbb{R} \))
- **Range**: All real numbers (\( \mathbb{R} \))
If you have graphing tools like Desmos, GeoGebra, or a graphing calculator, you can input the function \( f(x) = \sqrt[3]{x - 4} \) to see a visual representation.
Reviewed and approved by the UpStudy tutoring team
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