The function you've provided is:

\[
f(x) = \frac{x^2 + 3x - 4}{x^2 - 7x + 12}
\]

### Step 1: Factor the numerator and denominator
First, let's factor both the numerator and the denominator, if possible.

#### Numerator: \( x^2 + 3x - 4 \)
We need to factor this quadratic. We are looking for two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(x\)).

The factors of \(-4\) that sum to \(3\) are \(4\) and \(-1\). Thus, the numerator factors as:

\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]

#### Denominator: \( x^2 - 7x + 12 \)
Now, we factor the denominator. We are looking for two numbers that multiply to \(12\) (the constant term) and add to \(-7\) (the coefficient of \(x\)).

The factors of \(12\) that sum to \(-7\) are \(-3\) and \(-4\). Thus, the denominator factors as:

\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

### Step 2: Simplify the function
Now, the function becomes:

\[
f(x) = \frac{(x + 4)(x - 1)}{(x - 3)(x - 4)}
\]

Notice that the factor \((x - 4)\) appears in both the numerator and the denominator, so it can be canceled, but we must be careful to note that \(x \neq 4\), because division by zero is undefined.

Thus, the simplified function is:

\[
f(x) = \frac{x + 4}{x - 3}, \quad \text{for} \, x \neq 4
\]

### Step 3: Domain of the function
The function is undefined when the denominator is zero. Setting the original denominator equal to zero:

\[
x^2 - 7x + 12 = 0
\]
\[
(x - 3)(x - 4) = 0
\]

So, \(x = 3\) and \(x = 4\) are the points where the function is undefined. Therefore, the domain of the function is:

\[
x \in \mathbb{R}, \quad x \neq 3, 4
\]

### Final Answer:

\[
f(x) = \frac{x + 4}{x - 3}, \quad x \neq 3, 4
\]

Would you like to explore further details or any specific steps in this process?

Here are 5 related questions:
1. How would you find vertical asymptotes for this function?
2. What are the horizontal asymptotes of rational functions?
3. Can we determine the end behavior of \(f(x)\)?
4. How would you graph this function?
5. How can we determine the range of \(f(x)\)?

**Tip:** When simplifying rational functions, always check for values that make the denominator zero and exclude them from the domain.

This page provides a detailed solution for simplifying the rational function f(x) = (x² + 3x - 4)/(x² - 7x + 12). Learn how to factor quadratic expressions, simplify the function, and determine the domain. Suitable for 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