Task 3. Before working on this task 3, please read the following readings:

Read section 1.7 Transformations pages 120-139 of the following textbook will help you understand even and odd functions better.
Stitz, C., & Zeager, J. (2013). College algebra. Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdf 

If  f(x)= \sqrt[5]{x} , here are some transformations  \sqrt[5]{x}+6 ,   \sqrt[5]{x}-6 ,   \sqrt[5]{50x} ,   \sqrt[5]{x/50}  of  \sqrt[5]{x} 

Please answer the following questions:
(i) Draw the graphs of  f(x)= \sqrt[5]{x} ,   \sqrt[5]{x}+6 ,  \sqrt[5]{x}-6 ,  \sqrt[5]{50x} ,   \sqrt[5]{x/50}  of  \sqrt[5]{x} 

ii) Explain the above 4 transformations of  \sqrt[5]{x}  graphically.

iii) Write the observation on the domain and ranges of these 4 transformations of functions.

### (ii) Explanation of the transformations:

1. **\( f(x) = \sqrt[5]{x} \):**
   - This is the basic function. The graph passes through the origin, is increasing for both positive and negative values of \(x\), and has no vertical or horizontal shifts. It has a slight "flattening" compared to a square root function because it's a fifth root.

2. **\( f(x) = \sqrt[5]{x} + 6 \):**
   - **Vertical shift up** by 6 units. The entire graph of \( \sqrt[5]{x} \) moves up, meaning the range of the function increases by 6 units but the shape remains the same.

3. **\( f(x) = \sqrt[5]{x} - 6 \):**
   - **Vertical shift down** by 6 units. The graph of \( \sqrt[5]{x} \) is translated downward, decreasing the range by 6 units. The shape is preserved, but all values of \(f(x)\) are smaller.

4. **\( f(x) = \sqrt[5]{50x} \):**
   - **Horizontal compression** by a factor of 50. The function's growth becomes faster since multiplying \(x\) by 50 compresses the graph horizontally. This means that the graph looks steeper than the original.

5. **\( f(x) = \sqrt[5]{\frac{x}{50}} \):**
   - **Horizontal stretch** by a factor of 50. Dividing \(x\) by 50 stretches the graph horizontally, making it grow more slowly, thus the graph becomes flatter compared to the original.

### (iii) Domain and Range Observations:

- The **domain** for all these transformations is the same: \( (-\infty, \infty) \), because the fifth root function is defined for all real numbers.
  
- The **range** for the transformations:
  1. \( \sqrt[5]{x} \): Range is \( (-\infty, \infty) \).
  2. \( \sqrt[5]{x} + 6 \): Range shifts up by 6, still \( (-\infty, \infty) \).
  3. \( \sqrt[5]{x} - 6 \): Range shifts down by 6, still \( (-\infty, \infty) \).
  4. \( \sqrt[5]{50x} \): No vertical shift, so the range remains \( (-\infty, \infty) \).
  5. \( \sqrt[5]{\frac{x}{50}} \): No vertical shift, so the range remains \( (-\infty, \infty) \).

The primary effect of the transformations is on the steepness and vertical positioning of the graph, but the domain and range remain unchanged.

Would you like any further clarifications or details?

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**5 Related Questions:**
1. What would happen if we horizontally shift \( \sqrt[5]{x} \)?
2. How do horizontal and vertical shifts affect the domain and range?
3. Can the fifth root function ever have a limited domain?
4. What is the effect of combining both horizontal and vertical shifts?
5. How do different root functions (like cube root vs fifth root) compare graphically?

**Tip:** Horizontal transformations affect the \(x\)-coordinate (stretch/compression), while vertical transformations change the \(y\)-coordinate (shifts).

Task 3. Before working on this task 3, please read the following readings:

Read section 1.7 Transformations pages 120-139 of the following textbook will help you understand even and odd functions better.
Stitz, C., & Zeager, J. (2013). College algebra. Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdf 

If f(x) = √[5]{x}, here are some transformations √[5]{x}+6, √[5]{x}-6, √[5]{50x}, √[5]{x/50} of √[5]{x}

Please answer the following questions:
(i) Draw the graphs of f(x) = √[5]{x}, √[5]{x}+6, √[5]{x}-6, √[5]{50x}, √[5]{x/50} of √[5]{x}
(ii) Explain the above 4 transformations of √[5]{x} graphically.
(iii) Write the observation on the domain and ranges of these 4 transformations of functions.

Explore the graphical transformations of the fifth root function f(x) = √[5]{x}, including vertical shifts and horizontal stretches/compressions. Learn how these affect the graph, domain, and range of the function. Suitable for high school students (Grades 10-12).

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