Let me introduce some definitions beforehand.
Definition Let $X,Y \subset R ^n$ be disjoint sets. Suppose that $X'\subset X$ and $Y' \subset Y$, and $h: X' \rightarrow Y'$ is a homeomorphism. Define a partition $P(h)$ on $X \cup Y$ to be the collection of all pairs $\{ x, h(x)\}$ for $x \in X' $, and all single-element sets $\{z\}$ for $z \ in (X-X')\cup(Y-Y')$. A set $W \subset R^m$ is the result of attaching $X$ and $Y$ via the map $h$, denoted $X \cup _ h Y$, if $W$ is an identification space of $X \cup Y$ and $P(h)$.
Identification spaces are quotient spaces by the way.
Here is the question: Let $X = [0,2]$ and $Y=[3,5]$, let $X'=[1,2]$ and $Y'=[3,4]$, and let $h:X' \rightarrow Y'$ given by $h(x)=x+2$. Now, we are asked to find the quotient mapping and quotient space.
Here is my attempt:
Pictorially, we will find a quotient map such that the parts with red arrows will go to the same place. Geometrically, if we glue these parts together, we will get somewhat conic shape -not exactly, very roughly-. I am open to any help. Thanks in advance.
Define a continuous $f:[0,2] \cup [3,5] \to [0,3]$ by
$$f(x) = \begin{cases} x & 0 \le x \le 2\\ x - 2 & 3 \le x \le 5\\ \end{cases}$$
Then the induced map by $f$ to the quotient $X \cup_h Y$ (i.e. $\tilde{f}([x]) = f(x)$ for the classes $[x]$ in $X \cup_h Y$) is well-defined, 1-1 and onto and continuous so we have a homeomorphism with $[0,3]$.