Consider the rational function *f* given below.

*f*(*x*) = 

*x*3 + 8*x*2 + 15*x*

*x*2 − 3*x* − 18
 
Find the domain of *f*.

To determine the domain of the rational function \( f(x) = \frac{x^3 + 8x^2 + 15x}{x^2 - 3x - 18} \), we need to identify the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is equal to zero because division by zero is not allowed.

### Step 1: Find where the denominator equals zero
The denominator of the function is \( x^2 - 3x - 18 \). To find where this expression equals zero, we solve the equation:

\[
x^2 - 3x - 18 = 0
\]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:

\[
x^2 - 3x - 18 = (x - 6)(x + 3)
\]

### Step 3: Solve for \( x \)
Set each factor equal to zero:

\[
x - 6 = 0 \quad \text{or} \quad x + 3 = 0
\]

\[
x = 6 \quad \text{or} \quad x = -3
\]

### Step 4: Determine the domain
The function is undefined at \( x = 6 \) and \( x = -3 \) because these values make the denominator zero. Therefore, the domain of the function \( f(x) \) is all real numbers except \( x = 6 \) and \( x = -3 \).

### Final Answer:
The domain of \( f(x) \) is:
\[
\text{Domain of } f(x): \, \{ x \in \mathbb{R} \mid x \neq 6 \text{ and } x \neq -3 \}
\]

Or, in interval notation:
\[
(-\infty, -3) \cup (-3, 6) \cup (6, \infty)
\]

Would you like further details or have any questions?

### Related Questions:
1. How would you determine the range of this rational function?
2. Can you find the vertical asymptotes of the function?
3. What are the x-intercepts of the function?
4. How would you find the horizontal asymptote, if any?
5. Can you graph the function to visually confirm the domain and any asymptotes?

### Tip:
Always factor the denominator first to identify any restrictions in the domain of a rational function.

Discover how to find the domain of the rational function f(x) = (x^3 + 8x^2 + 15x) / (x^2 - 3x - 18) with a detailed step-by-step solution. This problem involves identifying where the function is defined and understanding the concept of rational functions and quadratic equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation