https://file.aiquickdraw.com/mathbot/file/1726142135027x8wyqbvg.jpg

The given function from the image is:

\[
f(x) = \frac{x^2 - 3x + 2}{x^2 - 4}
\]

This is a rational function, meaning it is a ratio of two polynomials. Let's simplify it if possible and discuss its properties, such as domain restrictions.

### Step 1: Factorize the numerator and denominator
- **Numerator:** \( x^2 - 3x + 2 \)
  - This factors as \( (x - 1)(x - 2) \).

- **Denominator:** \( x^2 - 4 \)
  - This is a difference of squares, which factors as \( (x - 2)(x + 2) \).

So, the function becomes:
\[
f(x) = \frac{(x - 1)(x - 2)}{(x - 2)(x + 2)}
\]

### Step 2: Cancel common factors
The term \( (x - 2) \) appears in both the numerator and the denominator, so we can cancel it out, provided \( x \neq 2 \) (because division by zero is undefined).

This simplifies to:
\[
f(x) = \frac{x - 1}{x + 2}, \quad \text{for} \ x \neq 2
\]

### Step 3: Domain restrictions
We have two values where the denominator of the original function becomes zero:
- \( x = 2 \)
- \( x = -2 \)

Thus, the domain of the function is all real numbers except \( x = 2 \) and \( x = -2 \).

### Final simplified form:
\[
f(x) = \frac{x - 1}{x + 2}, \quad \text{with} \ x \neq 2, \ x \neq -2
\]

Would you like to explore this function further, such as finding vertical and horizontal asymptotes or graphing the function? Let me know if you have any questions!

---

Here are 5 relative questions based on the function simplification:
1. What are the vertical asymptotes of the simplified function?
2. What is the horizontal asymptote of this rational function?
3. How would you find the x-intercept and y-intercept of this function?
4. Can you explain why \( x = 2 \) is a removable discontinuity?
5. How would you describe the behavior of this function as \( x \to \infty \)?

**Tip:** Always remember to check for domain restrictions before simplifying any rational expression to avoid undefined values.

Simplify the function f(x) = (x^2 - 3x + 2) / (x^2 - 4) and discuss its properties.

Learn how to simplify the rational function f(x) = (x^2 - 3x + 2) / (x^2 - 4) by factoring both the numerator and denominator. Discover key concepts like domain restrictions and removable discontinuities, suitable for high school math students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation