The image contains mathematical questions in Persian. I will analyze and solve the visible problem in the image. Here's a breakdown:

Question 10:

• A function $f(x)$ is defined with two cases: \begin{cases} x - 1 & \text{if } x < 2 \\ 2 - x^2 & \text{if } x \geq 2 \end{cases}$$

1. Graph the function $f(x)$.
2. Find the domain and range of $f(x)$.
3. Calculate $f(-1)$ and $f(1 + \sqrt{3})$.

Solutions:

1. Graph the Function $f(x)$:

• Case 1: When $x < 2$, $f(x) = x - 1$, which is a linear function with a slope of 1 and a y-intercept of -1.
• Case 2: When $x \geq 2$, $f(x) = 2 - x^2$, which is a downward parabola with a vertex at $(0, 2)$.

By combining these pieces, we can draw the function with the following points and conditions:

• For $x < 2$, graph the line $y = x - 1$.
• For $x \geq 2$, graph the parabola $y = 2 - x^2$.

2. Domain and Range:

• Domain: The function is defined for all real numbers $x$. Hence, the domain is: $\text{Domain} = (-\infty, \infty)$

• Range: The line $y = x - 1$ for $x < 2$ covers $(-\infty, 1)$. The parabola $y = 2 - x^2$ for $x \geq 2$ has a vertex at $(2, -2)$ and decreases infinitely. Thus, the overall range is: $\text{Range} = (-\infty, 1) \cup [-2, 2]$

3. Calculate $f(-1)$ and $f(1 + \sqrt{3})$:

• $f(-1)$: Since $-1 < 2$, use the formula $f(x) = x - 1$: $f(-1) = -1 - 1 = -2$

• $f(1 + \sqrt{3})$: Since $1 + \sqrt{3} \approx 2.732$ and $2.732 > 2$, use the formula $f(x) = 2 - x^2$: $f(1 + \sqrt{3}) = 2 - (1 + \sqrt{3})^2$ Expand $(1 + \sqrt{3})^2$: $(1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}$ So: $f(1 + \sqrt{3}) = 2 - (4 + 2\sqrt{3}) = -2 - 2\sqrt{3}$

Final Results:

1. Graph: Constructed by combining the line $y = x - 1$ for $x < 2$ and the parabola $y = 2 - x^2$ for $x \geq 2$.
2. Domain: $(-\infty, \infty)$.
3. Range: $(-\infty, 1) \cup [-2, 2]$.
4. $f(-1) = -2$.
5. $f(1 + \sqrt{3}) = -2 - 2\sqrt{3}$.

Would you like me to graph the function or explain any specific part in more detail?

Related Questions:

1. What is the difference between piecewise-defined functions and continuous functions?
2. How do you determine the range of a quadratic function graphically?
3. What is the effect of $x < c$ and $x \geq c$ conditions in piecewise functions?
4. How do you calculate function values for irrational arguments like $\sqrt{3}$?
5. How would the graph and range change if $f(x)$ were modified to $f(x) = x^2 - 2x + 1$ for $x < 2$?

Tip:

Always carefully evaluate the conditions for piecewise functions to ensure the correct formula is applied!