Graph the function $$ f(x) = \sqrt[3]{x - 4} $$ and identify its domain and range.

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UpStudy AI · Accepted Answer

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.
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

Graph the function \( f(x) = \sqrt[3]{x - 4} \) and identify its domain and range.

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