https://file.aiquickdraw.com/mathbot/file/1726021222533vw1c7p00.jpg

Let's solve the problems from the image. Each problem asks you to find the domain and range of a given rational function.

### Independent Practice

1. **\( f(x) = \frac{2}{x+1} \)**

   - **Domain:** \( x \neq -1 \) (since \( x + 1 = 0 \) when \( x = -1 \))
   - **Range:** \( y \neq 0 \) (as \( \frac{2}{x+1} = 0 \) has no solution)

2. **\( f(x) = \frac{3x}{x+3} \)**

   - **Domain:** \( x \neq -3 \) (since \( x + 3 = 0 \) when \( x = -3 \))
   - **Range:** \( y \neq 3 \) (because the horizontal asymptote, as \( x \) approaches infinity, is \( y = 3 \))

3. **\( f(x) = \frac{3-x}{x-7} \)**

   - **Domain:** \( x \neq 7 \) (since \( x - 7 = 0 \) when \( x = 7 \))
   - **Range:** \( y \neq -1 \) (since the function approaches -1 as \( x \) increases)

4. **\( f(x) = \frac{2+x}{x} \)**

   - **Domain:** \( x \neq 0 \) (since the denominator cannot be zero)
   - **Range:** \( y \neq 1 \) (since the function \( \frac{2+x}{x} \) simplifies to \( 1 + \frac{2}{x} \))

5. **\( f(x) = \frac{1}{x^2 - 1} \)**

   - **Domain:** \( x \neq 1 \) and \( x \neq -1 \) (since \( x^2 - 1 = 0 \) at \( x = \pm 1 \))
   - **Range:** \( y > 0 \) (since the function is always positive and cannot equal zero)

### Independent Assessment

1. **\( f(x) = \frac{3}{x^2 - 1} \)**

   - **Domain:** \( x \neq 1 \) and \( x \neq -1 \)
   - **Range:** \( y > 0 \)

2. **\( f(x) = \frac{2x}{x-4} \)**

   - **Domain:** \( x \neq 4 \)
   - **Range:** \( y \neq 2 \)

3. **\( f(x) = \frac{x + 3}{5x - 5} \)**

   - **Domain:** \( x \neq 1 \)
   - **Range:** \( y \neq \frac{2}{5} \)

4. **\( f(x) = \frac{2+x}{2x} \)**

   - **Domain:** \( x \neq 0 \)
   - **Range:** \( y \neq \frac{1}{2} \)

5. **\( f(x) = \frac{x^2 + 4x + 3}{x^2 - 9} \)**

   - **Domain:** \( x \neq 3 \) and \( x \neq -3 \)
   - **Range:** \( y \neq 1 \)

Would you like more details on any of these steps, or do you have any questions?

### Relative Questions:
1. What happens to the range of a rational function when the numerator is constant?
2. How do you find vertical asymptotes of a rational function?
3. Why does the function \( f(x) = \frac{2}{x+1} \) never equal zero?
4. What is the relationship between horizontal asymptotes and the range of a rational function?
5. How can you determine the domain of a rational function by looking at its equation?

### Tip:
When analyzing the range of a rational function, consider the function's behavior as \( x \) approaches very large positive or negative values (horizontal asymptotes) and also check for values that make the numerator zero, as these can help identify values the function cannot take.

Find the domain and range of the following rational functions.

Learn how to find the domain and range of rational functions. This problem set covers key algebraic concepts like identifying vertical and horizontal asymptotes to determine the range and avoiding zero denominators for the domain. Suitable for Grades 10-12 students.

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