https://file.aiquickdraw.com/mathbot/file/1728047077687pshoogep.jpg

Let's break down the mathematical tasks from the image:

### 1. Function \( f(x) = \sqrt{x^2 - 10} + 1 \)
   - **(a)** Determine the domain and range.
   - **(b)** Explain any restrictions on the domain and range.
   - **(c)** Draw the graph of the function.

### 2. Given two functions \( f(x) = x + 2 \) and \( g(x) = \frac{1}{x - 3} \), perform:
   - **(a)** Find \( (f + g)(x) \) and simplify.
   - **(b)** Find \( (f \circ g)(x) \) and simplify.
   - **(c)** Find \( (g \circ f)(x) \) and simplify.
   - **(d)** Determine the domain and range of the resulting functions from parts (a), (b), and (c).

### 3. Consider the line passing through points \( A(3, -1) \) and \( B(-2, 4) \):
   - **(a)** Calculate the slope.
   - **(b)** Write the equation in point-slope form.
   - **(c)** Convert from point-slope form to slope-intercept form.
   - **(d)** Verify the intercepts by graphing the line.

### 4. Graph the piecewise function \( f(x) \):
   \[
   f(x) = 
   \begin{cases}
   |x + 1| & x < -5 \\
   (x - 1)^2 + 1 & -5 \leq x < 5 \\
   |x + 1| & x \geq 5
   \end{cases}
   \]

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Solve various mathematical problems involving domain and range, function composition, linear equations, and piecewise functions.

Explore how to determine the domain and range, perform function compositions, solve linear equations, and graph piecewise functions. Ideal for high school students, this guide covers concepts like domain restrictions, slope-intercept form, and properties of functions.

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