https://file.aiquickdraw.com/mathbot/file/1729825189521cszyhj2a.jpg

Let's analyze the problem step-by-step.

### Function Analysis
Given the function:
\[
f(x) = \frac{2x^2 - 8}{x^2 - 6x + 8}
\]

We need to find:
1. **x-intercepts** (where \( f(x) = 0 \)),
2. **Vertical asymptotes** (values of \( x \) where the denominator is 0),
3. **Holes** (common factors in numerator and denominator).

### Step 1: **Simplify the Function**
#### Factor the Numerator and Denominator
**Numerator:**
\[
2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)
\]

**Denominator:**
\[
x^2 - 6x + 8 = (x - 2)(x - 4)
\]

So the function can be rewritten as:
\[
f(x) = \frac{2(x - 2)(x + 2)}{(x - 2)(x - 4)}
\]

### Step 2: **Identify Holes and Vertical Asymptotes**
- There is a common factor of \( (x - 2) \) in both the numerator and the denominator.
- **Hole** at \( x = 2 \), since this factor cancels out.
- After canceling, the simplified function is:
\[
f(x) = \frac{2(x + 2)}{x - 4}
\]
- **Vertical Asymptote** is where the denominator is 0 after simplification:
  \[
  x - 4 = 0 \implies x = 4
  \]

### Step 3: **Find x-intercepts**
To find x-intercepts, set the numerator of the simplified function to zero:
\[
2(x + 2) = 0
\]
\[
x + 2 = 0 \implies x = -2
\]

### Summary
1. **x-intercept**: \( x = -2 \).
2. **Vertical asymptote**: \( x = 4 \).
3. **Hole**: At \( x = 2 \).

Would you like more details or have any questions?

### Related Questions
1. How do we determine horizontal asymptotes for this function?
2. What happens to the graph of \( f(x) \) near the hole at \( x = 2 \)?
3. Can you explain why there is no horizontal asymptote in this case?
4. How would the function behave if there were no common factors between the numerator and denominator?
5. How do we find slant (oblique) asymptotes if they exist?

### Tip:
To check for holes, always look for common factors in the numerator and denominator before simplifying.

Find any x-intercepts, vertical asymptotes, and holes for the graph of the following function: f(x) = (2x^2 - 8) / (x^2 - 6x + 8)

Learn how to find x-intercepts, vertical asymptotes, and holes for the rational function f(x) = (2x^2 - 8) / (x^2 - 6x + 8). This problem covers factoring, identifying asymptotes, and canceling common factors. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation