https://file.aiquickdraw.com/mathbot/file/1727756782951j6n5sq69.jpg

The image presents two parts, labeled (a) and (b), that require analyzing two different functions to complete specific statements. Let’s break down each part.

### Part (a)
The function \( f \) is defined as:
\[
f : \mathbb{R}^+_0 \to \mathbb{R}, \quad f(x) = x(x + 1)^2
\]
- The function maps from the set of non-negative real numbers \( \mathbb{R}^+_0 \) to the real numbers \( \mathbb{R} \).
- You need to identify the characteristics of the equation \( f(x) = x(x + 1)^2 \), such as its behavior or properties like injectivity or surjectivity, in order to complete the statement.

We need to analyze whether \( f(x) \) has a specific property (e.g., injectivity, surjectivity, or both) and then conclude whether \( f \) is injective, surjective, or bijective.

### Part (b)
The function \( g \) is defined as:
\[
g : \mathbb{R}^+_0 \to \mathbb{R}^+_0, \quad g(x) = (x - 4)^2
\]
- The function maps from the set of non-negative real numbers to itself.
- Similarly to part (a), the statement requires determining the properties of \( g(x) = (x - 4)^2 \) and concluding whether \( g \) is injective, surjective, or both.

### Steps to Solve:
1. **Analyze Function \( f(x) = x(x + 1)^2 \) (Part a):**
   - Investigate injectivity: Check whether different inputs \( x_1 \) and \( x_2 \) can give the same output.
   - Investigate surjectivity: Determine if every value in the target space \( \mathbb{R} \) has a pre-image in the domain.

2. **Analyze Function \( g(x) = (x - 4)^2 \) (Part b):**
   - For injectivity: Determine if distinct values of \( x \) lead to distinct outputs.
   - For surjectivity: Check whether every non-negative real number has a pre-image.

Let me know if you would like a detailed breakdown of these analyses! Would you also like to know more about the behavior of these types of functions in general?

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Here are five questions you can explore further:
1. What is the definition of an injective function?
2. How can you determine if a function is surjective?
3. What is the geometric interpretation of \( f(x) = x(x + 1)^2 \)?
4. How does the function \( g(x) = (x - 4)^2 \) behave for values of \( x \) close to 4?
5. What is the relationship between injectivity, surjectivity, and bijectivity?

### Tip:
When determining injectivity, try plugging in two different values for \( x \) and see if their outputs can ever be equal.

Analyze the functions f(x) = x(x + 1)^2 and g(x) = (x - 4)^2 to determine their injectivity and surjectivity.

Explore the properties of injectivity and surjectivity for the functions f(x) = x(x + 1)^2 and g(x) = (x - 4)^2. This detailed explanation helps students in Grades 10-12 understand key concepts related to quadratic functions and their behavior.

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