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.