The image contains a portion of a mathematical exercise from a textbook. Based on the visible text, it appears to be related to the properties of functions, particularly bijections, and function compositions.

The text reads:

• Exercise 3.3.12. Suppose $f: X \to Y$ and $g: Y \to Z$, and let $g \circ f = Id_X$ (identity function on $X$).
  • Show that $f$ is one-to-one (injective) and $g$ is onto (surjective).

This is a typical problem involving function composition and properties like injectivity (one-to-one) and surjectivity (onto). Let's break down the task of proving the statements.

Problem Breakdown:

You are given two functions $f: X \to Y$ and $g: Y \to Z$, such that their composition $g \circ f$ is the identity function on $X$, denoted by $Id_X$. This means for all $x \in X$, $g(f(x)) = x$.

• To show:
  1. $f$ is injective (one-to-one).
  2. $g$ is surjective (onto).

Hints for Solution:

1. Proving that $f$ is injective:

   • If $f(x_1) = f(x_2)$ for $x_1, x_2 \in X$, apply $g$ to both sides. You will use the fact that $g \circ f = Id_X$ to demonstrate that $x_1 = x_2$, which implies that $f$ is injective.

2. Proving that $g$ is surjective:

   • Since $g \circ f = Id_X$, for every $x \in X$, there is a corresponding $f(x) \in Y$, such that $g(f(x)) = x$. This suggests that $g$ maps some element of $Y$ to every element of $X$, helping establish that $g$ is surjective.

Let me know if you'd like a detailed step-by-step solution, or if you have any specific questions!

5 Related Questions:

1. What is the difference between injective and surjective functions?
2. How can you prove that a function is injective?
3. What is the significance of the identity function in function composition?
4. Can a function be bijective if it is not injective or surjective?
5. How do function compositions affect the properties of injectivity and surjectivity?

Tip: When working with function compositions, always keep in mind how the properties of the individual functions affect the overall composition.