Question
Sketch the graph of the following rational functions and identify the x intercept, y intercept of it exist and vertical asymptote and horizontal asymptote. Also stated the domain and range of the function.
A. Y=x_1÷x²_9
B. Y=square root x+4
A. Y=x_1÷x²_9
B. Y=square root x+4
Answers
GPT-4o mini
Let's analyze and sketch the provided functions as requested.
### A. \( y = \frac{x}{x^2 - 9} \)
**1. x-intercept:**
- Set \( y = 0 \):
\[
\frac{x}{x^2 - 9} = 0 \implies x = 0
\]
So the x-intercept is \( (0, 0) \).
**2. y-intercept:**
- Set \( x = 0 \):
\[
y = \frac{0}{0^2 - 9} = 0
\]
So the y-intercept is also \( (0, 0) \).
**3. Vertical asymptotes:**
- Find values where the denominator is zero:
\[
x^2 - 9 = 0 \implies x^2 = 9 \implies x = 3 \quad \text{and} \quad x = -3
\]
Vertical asymptotes are at \( x = 3 \) and \( x = -3 \).
**4. Horizontal asymptote:**
- As \( x \to \infty \) or \( x \to -\infty \), the dominant term in the numerator and denominator is \( x \) and \( x^2 \), respectively. Thus:
\[
y \approx \frac{x}{x^2} = \frac{1}{x} \to 0
\]
The horizontal asymptote is \( y = 0 \).
**5. Domain:**
- The function is undefined where the denominator is zero:
\[
x^2 - 9 = 0 \implies x \neq -3, 3
\]
The domain is \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \).
**6. Range:**
- The horizontal asymptote at \( y = 0 \) indicates that the function will approach \( 0 \) but not reach it. The function goes off to \( \pm \infty \) near the vertical asymptotes. Therefore, the range is \( (-\infty, 0) \cup (0, \infty) \).
### Sketch:
- The graph will cross the x-axis at \( (0, 0) \), will have vertical asymptotes at \( x = -3 \) and \( x = 3 \), and a horizontal asymptote at \( y = 0 \). The graph approaches but never reaches \( y = 0 \).
---
### B. \( y = \sqrt{x + 4} \)
**1. x-intercept:**
- Set \( y = 0 \):
\[
\sqrt{x + 4} = 0 \implies x + 4 = 0 \implies x = -4
\]
So the x-intercept is \( (-4, 0) \).
**2. y-intercept:**
- Set \( x = 0 \):
\[
y = \sqrt{0 + 4} = \sqrt{4} = 2
\]
So the y-intercept is \( (0, 2) \).
**3. Vertical asymptotes:**
- There are no vertical asymptotes for the square root function.
**4. Horizontal asymptote:**
- As \( x \to \infty \), \( y \) continues to increase without bound. Thus, there is no horizontal asymptote.
**5. Domain:**
- The expression inside the square root must be non-negative:
\[
x + 4 \geq 0 \implies x \geq -4
\]
The domain is \( [-4, \infty) \).
**6. Range:**
- The output \( y \) can never be negative (since a square root is always non-negative):
\[
y \geq 0
\]
The range is \( [0, \infty) \).
### Sketch:
- The graph starts at the point \( (-4, 0) \) and curves upward to the right. The y-intercept at \( (0, 2) \) is visible, and it continues indefinitely in the positive y-direction, but does not have any vertical or horizontal asymptotes.
---
### Summary:
**A. \( y = \frac{x}{x^2 - 9} \)**
- x-intercept: \( (0, 0) \)
- y-intercept: \( (0, 0) \)
- Vertical Asymptotes: \( x = -3, \, x = 3 \)
- Horizontal Asymptote: \( y = 0 \)
- Domain: \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \)
- Range: \( (-\infty, 0) \cup (0, \infty) \)
**B. \( y = \sqrt{x + 4} \)**
- x-intercept: \( (-4, 0) \)
- y-intercept: \( (0, 2) \)
- Vertical Asymptote: None
- Horizontal Asymptote: None
- Domain: \( [-4, \infty) \)
- Range: \( [0, \infty) \)
### A. \( y = \frac{x}{x^2 - 9} \)
**1. x-intercept:**
- Set \( y = 0 \):
\[
\frac{x}{x^2 - 9} = 0 \implies x = 0
\]
So the x-intercept is \( (0, 0) \).
**2. y-intercept:**
- Set \( x = 0 \):
\[
y = \frac{0}{0^2 - 9} = 0
\]
So the y-intercept is also \( (0, 0) \).
**3. Vertical asymptotes:**
- Find values where the denominator is zero:
\[
x^2 - 9 = 0 \implies x^2 = 9 \implies x = 3 \quad \text{and} \quad x = -3
\]
Vertical asymptotes are at \( x = 3 \) and \( x = -3 \).
**4. Horizontal asymptote:**
- As \( x \to \infty \) or \( x \to -\infty \), the dominant term in the numerator and denominator is \( x \) and \( x^2 \), respectively. Thus:
\[
y \approx \frac{x}{x^2} = \frac{1}{x} \to 0
\]
The horizontal asymptote is \( y = 0 \).
**5. Domain:**
- The function is undefined where the denominator is zero:
\[
x^2 - 9 = 0 \implies x \neq -3, 3
\]
The domain is \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \).
**6. Range:**
- The horizontal asymptote at \( y = 0 \) indicates that the function will approach \( 0 \) but not reach it. The function goes off to \( \pm \infty \) near the vertical asymptotes. Therefore, the range is \( (-\infty, 0) \cup (0, \infty) \).
### Sketch:
- The graph will cross the x-axis at \( (0, 0) \), will have vertical asymptotes at \( x = -3 \) and \( x = 3 \), and a horizontal asymptote at \( y = 0 \). The graph approaches but never reaches \( y = 0 \).
---
### B. \( y = \sqrt{x + 4} \)
**1. x-intercept:**
- Set \( y = 0 \):
\[
\sqrt{x + 4} = 0 \implies x + 4 = 0 \implies x = -4
\]
So the x-intercept is \( (-4, 0) \).
**2. y-intercept:**
- Set \( x = 0 \):
\[
y = \sqrt{0 + 4} = \sqrt{4} = 2
\]
So the y-intercept is \( (0, 2) \).
**3. Vertical asymptotes:**
- There are no vertical asymptotes for the square root function.
**4. Horizontal asymptote:**
- As \( x \to \infty \), \( y \) continues to increase without bound. Thus, there is no horizontal asymptote.
**5. Domain:**
- The expression inside the square root must be non-negative:
\[
x + 4 \geq 0 \implies x \geq -4
\]
The domain is \( [-4, \infty) \).
**6. Range:**
- The output \( y \) can never be negative (since a square root is always non-negative):
\[
y \geq 0
\]
The range is \( [0, \infty) \).
### Sketch:
- The graph starts at the point \( (-4, 0) \) and curves upward to the right. The y-intercept at \( (0, 2) \) is visible, and it continues indefinitely in the positive y-direction, but does not have any vertical or horizontal asymptotes.
---
### Summary:
**A. \( y = \frac{x}{x^2 - 9} \)**
- x-intercept: \( (0, 0) \)
- y-intercept: \( (0, 0) \)
- Vertical Asymptotes: \( x = -3, \, x = 3 \)
- Horizontal Asymptote: \( y = 0 \)
- Domain: \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \)
- Range: \( (-\infty, 0) \cup (0, \infty) \)
**B. \( y = \sqrt{x + 4} \)**
- x-intercept: \( (-4, 0) \)
- y-intercept: \( (0, 2) \)
- Vertical Asymptote: None
- Horizontal Asymptote: None
- Domain: \( [-4, \infty) \)
- Range: \( [0, \infty) \)