1) \( f(x)=\frac{x}{x+3} \)

The function you provided is \[ f(x) = \frac{x}{x + 3}. \] Here are some key features of this function: 1. **Domain**: The function is defined for all real numbers except where the denominator is zero. To find the value that makes the denominator zero, set \( x + 3 = 0 \): \[ x = -3. \] Therefore, the domain of \( f(x) \) is \( x \in \mathbb{R}, x \neq -3 \). 2. **Vertical Asymptote**: Since the function is undefined at \( x = -3 \), there is a vertical asymptote at \( x = -3 \). 3. **Horizontal Asymptote**: To find the horizontal asymptote, we can analyze the behavior of \( f(x) \) as \( x \) approaches infinity or negative infinity. As \( x \to \infty \) or \( x \to -\infty \): \[ f(x) = \frac{x}{x + 3} \approx \frac{x}{x} = 1. \] Thus, there is a horizontal asymptote at \( y = 1 \). 4. **Intercepts**: - **x-intercept**: Set \( f(x) = 0 \): \[ \frac{x}{x + 3} = 0 \implies x = 0. \] So, the x-intercept is at \( (0, 0) \). - **y-intercept**: Evaluate \( f(0) \): \[ f(0) = \frac{0}{0 + 3} = 0. \] So, the y-intercept is also at \( (0, 0) \). 5. **Behavior near the asymptotes**: - As \( x \) approaches -3 from the left (\( x \to -3^- \)), \( f(x) \to \frac{-3}{0^-} \to +\infty \). - As \( x \) approaches -3 from the right (\( x \to -3^+ \)), \( f(x) \to \frac{-3}{0^+} \to -\infty \). 6. **Graph**: The graph of \( f(x) \) will show a hyperbolic shape with the vertical asymptote at \( x = -3 \) and the horizontal asymptote at \( y = 1 \). If you have any specific questions or need further analysis on this function, feel free to ask!