The function given in the problem is $f: D \to C$ defined as $f(x) = |2x - 3|$, with $D, C \subset \mathbb{R}$. Let's solve each part step-by-step:

(a) Find examples for when $f$ is injective and not injective.

Injective Example:

A function $f$ is injective (one-to-one) if each value in the codomain $C$ is mapped by at most one value in the domain $D$.

Consider the function $f(x) = |2x - 3|$. To make this function injective, we need to avoid situations where two different values of $x$ could give the same output. Since the absolute value function $| \cdot |$ is symmetric, it will not be injective over the entire real line $\mathbb{R}$. However, we can restrict the domain to one side of the symmetry.

Example of an injective case:

• Let $D = \left[\frac{3}{2}, \infty \right)$, meaning $x \geq \frac{3}{2}$.
• Let $C = [0, \infty)$, because $f(x) = |2x - 3| \geq 0$ for all $x$.

In this case, $f(x) = 2x - 3$, which is a linear function on this interval, and linear functions are injective. So, $f$ is injective.

Non-Injective Example:

For the function to be non-injective, we need the domain to allow for two different values of $x$ that give the same output.

Example of a non-injective case:

• Let $D = \mathbb{R}$ (the entire real line).
• Let $C = [0, \infty)$.

For this domain, the function $f(x) = |2x - 3|$ is not injective because, for example:

• $f(0) = |2(0) - 3| = 3$,
• $f(3) = |2(3) - 3| = 3$.

Thus, $f(0) = f(3) = 3$, but $0 \neq 3$, which shows that the function is not injective.

(b) Find examples for when $f$ is surjective and not surjective.

Surjective Example:

A function $f$ is surjective (onto) if every element in the codomain $C$ is mapped by at least one value in the domain $D$.

Example of a surjective case:

• Let $D = \mathbb{R}$ (the entire real line).
• Let $C = [0, \infty)$.

For every $y \in [0, \infty)$, there is some $x \in \mathbb{R}$ such that $f(x) = y$. Specifically, solve $|2x - 3| = y$ for $x$:

• If $y \geq 0$, then $2x - 3 = \pm y$, which gives solutions for $x$. Thus, every value of $y \in C$ has a preimage, and the function is surjective.

Non-Surjective Example:

For the function to be non-surjective, we need to restrict the codomain so that not every value in the codomain is reached.

Example of a non-surjective case:

• Let $D = \mathbb{R}$.
• Let $C = (1, \infty)$.

Here, the function is not surjective because values in $[0, 1]$ are never achieved by the function $f(x) = |2x - 3|$. For example, there is no $x \in \mathbb{R}$ such that $f(x) = 0.5$.

Let me know if you would like more details or have any further questions!

Here are some additional related questions:

1. Can we generalize the function $f(x) = |ax + b|$ for injectivity and surjectivity?
2. What are the implications of restricting $D$ or $C$ on the range of a function?
3. How can we determine if a piecewise function is injective or surjective?
4. How does symmetry affect the injectivity of functions like $|2x - 3|$?
5. What other transformations on $x$ would impact the surjectivity of the absolute value function?

Tip: To test injectivity, use the horizontal line test on the graph of the function: If any horizontal line crosses the graph more than once, the function is not injective.