Find the domain of the rational function : f(x) = 3x(x-1)/2x^2 -5x-12

To find the domain of the rational function

\[
f(x) = \frac{3x(x - 1)}{2x^2 - 5x - 12}
\]

we need to determine the values of \( x \) for which the function is defined. Since it's a rational function, the denominator must not be zero. So, we solve for the values of \( x \) that make the denominator zero and exclude these values from the domain.

### Step 1: Set the Denominator to Zero
The denominator of \( f(x) \) is \( 2x^2 - 5x - 12 \). Set it equal to zero and solve for \( x \):

\[
2x^2 - 5x - 12 = 0
\]

### Step 2: Factor the Quadratic Equation
To factor \( 2x^2 - 5x - 12 \), we look for two numbers that multiply to \(-24\) (the product of \(2\) and \(-12\)) and add up to \(-5\) (the middle coefficient).

These numbers are \( -8 \) and \( 3 \), so we can rewrite the expression as:

\[
2x^2 - 8x + 3x - 12 = 0
\]

Now, factor by grouping:

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

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

### Step 3: Solve for \( x \)
Now, set each factor to zero:

1. \( 2x + 3 = 0 \) gives \( x = -\frac{3}{2} \)
2. \( x - 4 = 0 \) gives \( x = 4 \)

### Step 4: Determine the Domain
Since \( f(x) \) is undefined at \( x = -\frac{3}{2} \) and \( x = 4 \), we exclude these values from the domain.

Thus, the domain of \( f(x) \) is all real numbers except \( x = -\frac{3}{2} \) and \( x = 4 \). In interval notation, this is:

\[
\text{Domain of } f(x) = (-\infty, -\frac{3}{2}) \cup \left(-\frac{3}{2}, 4\right) \cup (4, \infty)
\]

Would you like more details on any of these steps? 

Here are some related questions:
1. How do you find the domain of a function with both square roots and rational expressions?
2. What happens to the graph of a function at points excluded from its domain?
3. Can factoring always be used to find where a quadratic expression is zero?
4. How do you find the range of a rational function?
5. What are vertical asymptotes and how are they related to the domain?

**Tip:** Always set the denominator of a rational function equal to zero to identify points excluded from its domain.

Find the domain of the rational function: f(x) = 3x(x-1)/2x^2 - 5x - 12

Learn how to determine the domain of the rational function f(x) = 3x(x - 1) / (2x^2 - 5x - 12). This guide covers factoring the quadratic expression in the denominator, setting it to zero, and identifying values excluded from the domain. 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