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?

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.