Coles George
11/23/2023 · Middle School
1) \( f(x)=\frac{x}{x+3} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
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 (\( x \to -3^- \)), \( f(x) \to \infty \).
- As \( x \) approaches -3 from the right (\( x \to -3^+ \)), \( f(x) \to -\infty \).
6. **Graph**: The graph of the function 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!
Quick Answer
The function \( f(x) = \frac{x}{x + 3} \) has a vertical asymptote at \( x = -3 \), a horizontal asymptote at \( y = 1 \), and intercepts at \( (0, 0) \).
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