1. **Stating the problem:** We want to understand the general concept of functions, especially for beginners in 1st year Baccalaureate (1bac) in Morocco, explained in English with detailed examples. 2. **Definition of a function:** A function is a rule that assigns to each element $x$ in a set called the domain exactly one element $y$ in another set called the codomain. We write this as $y = f(x)$, where $f$ is the function. 3. **Example 1:** Consider the function $f(x) = 2x + 3$. For each input $x$, the output is calculated by doubling $x$ and then adding 3. 4. **Evaluating example 1:** If $x = 1$, then $f(1) = 2 \times 1 + 3 = 5$. If $x = 4$, then $f(4) = 2 \times 4 + 3 = 11$. 5. **Example 2:** The function $g(x) = x^2$ assigns to each number its square. For example, $g(3) = 9$ and $g(-2) = 4$. 6. **Domain and codomain:** The domain is the set of all possible inputs. For $f(x) = 2x + 3$, the domain can be all real numbers $\mathbb{R}$. The codomain is also $\mathbb{R}$ because outputs can be any real number. 7. **Graph of a function:** The graph of $f(x) = 2x + 3$ is a straight line with slope 2 and y-intercept 3. The graph of $g(x) = x^2$ is a parabola opening upwards. 8. **Summary:** Functions link inputs to outputs with a clear rule. Understanding functions helps in many areas of math and science. Final answer: A function is a rule that assigns exactly one output to each input, written as $y = f(x)$, with examples like $f(x) = 2x + 3$ and $g(x) = x^2$ illustrating this concept clearly.