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Let's analyze the three given functions to determine their domain and range:

### 9. \( f(x) = x - 1 \)
- **Domain**: Since this is a linear function, it is defined for all real numbers. Hence, the domain is \( (-\infty, \infty) \).
- **Range**: The output (range) of a linear function is also all real numbers, so the range is \( (-\infty, \infty) \).

### 11. \( f(x) = 4 \), for \( 1 \leq x \leq 3 \)
- **Domain**: The domain is explicitly restricted by the given condition \( 1 \leq x \leq 3 \). So, the domain is \( [1, 3] \).
- **Range**: The function is a constant function, always equal to 4. Hence, the range is \( \{4\} \).

### 13. \( f(x) = 4 - x^2 \)
- **Domain**: This is a quadratic function, and since there are no restrictions on \( x \), the domain is all real numbers, i.e., \( (-\infty, \infty) \).
- **Range**: To find the range, notice that the function \( f(x) = 4 - x^2 \) is a downward-opening parabola with its vertex at \( (0, 4) \). The maximum value is 4, and as \( x \) increases or decreases, \( f(x) \) decreases. Therefore, the range is \( (-\infty, 4] \).

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Do you want more details or have any questions? Here are 5 related questions to explore:

1. How do we find the domain of more complex functions?
2. Why does a constant function like \( f(x) = 4 \) have only one value in its range?
3. What techniques can be used to find the range of a quadratic function like \( f(x) = 4 - x^2 \)?
4. How does the vertex of a parabola help in determining the range?
5. What are the differences between linear and quadratic functions in terms of their graphs?

**Tip:** Always look for any restrictions (like square roots, denominators, or explicit conditions) in a function to correctly identify the domain.

Se da una función f. (a) encuentra el dominio y rango de f a partir de la gráfica.
9. f(x) = x - 1
11. f(x) = 4, 1 ≤ x ≤ 3
13. f(x) = 4 - x²

Explore how to find the domain and range of various functions such as linear, constant, and quadratic. This problem includes the functions f(x) = x - 1, f(x) = 4 with a defined interval, and f(x) = 4 - x². Ideal for high school students studying algebra.

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