### Finding the Domain of the Rational FunctionTo determine the domain of the function \[ f(x) = \frac{x^2 - 36}{2x^2 + 7x - 15} \], we need to ensure that the denominator is not equal to zero, as division by zero is undefined. #### Steps to Find the Domain:1. **Identify the denominator**: The denominator of the function is \( 2x^2 + 7x - 15 \).2. **Set the denominator equal to zero**: \( 2x^2 + 7x - 15 = 0 \).3. **Solve the quadratic equation**: - Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 7 \), and \( c = -15 \). - Calculate the discriminant: \[ \Delta = b^2 - 4ac = 7^2 - 4(2)(-15) = 49 + 120 = 169 \]. - Find the roots: \[ x = \frac{-7 \pm \sqrt{169}}{2(2)} = \frac{-7 \pm 13}{4} \]. \[ x_1 = \frac{6}{4} = \frac{3}{2} \]. \[ x_2 = \frac{-20}{4} = -5 \].4. **Domain identification**: The domain of the function \( f(x) \) excludes the values \( x = \frac{3}{2} \) and \( x = -5 \). Therefore, \[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq \frac{3}{2} \text{ and } x \neq -5 \} \].In conclusion, the domain of the function \( f(x) = \frac{x^2 - 36}{2x^2 + 7x - 15} \) consists of all real numbers except \( x = \frac{3}{2} \) and \( x = -5 \).

Given: The rational function f(x)=x2-362x2+7x-15 Find the domain of the given function.

Answered: : f(x) = ebraically, find the domain of…

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: f(x) = ebraically, find the domain of x²-36 2x²+7x-15. Show your work for full credit

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Finding the Domain of the Rational Function

To determine the domain of the function \[ f(x) = \frac{x^2 - 36}{2x^2 + 7x - 15} \], we need to ensure that the denominator is not equal to zero, as division by zero is undefined.

#### Steps to Find the Domain:
1. **Identify the denominator**:
The denominator of the function is \( 2x^2 + 7x - 15 \).

2. **Set the denominator equal to zero**:
\( 2x^2 + 7x - 15 = 0 \).

3. **Solve the quadratic equation**:
- Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 7 \), and \( c = -15 \).
- Calculate the discriminant:
\[ \Delta = b^2 - 4ac = 7^2 - 4(2)(-15) = 49 + 120 = 169 \].
- Find the roots:
\[ x = \frac{-7 \pm \sqrt{169}}{2(2)} = \frac{-7 \pm 13}{4} \].
\[ x_1 = \frac{6}{4} = \frac{3}{2} \].
\[ x_2 = \frac{-20}{4} = -5 \].

4. **Domain identification**:
The domain of the function \( f(x) \) excludes the values \( x = \frac{3}{2} \) and \( x = -5 \). Therefore,

\[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq \frac{3}{2} \text{ and } x \neq -5 \} \].

In conclusion, the domain of the function \( f(x) = \frac{x^2 - 36}{2x^2 + 7x - 15} \) consists of all real numbers except \( x = \frac{3}{2} \) and \( x = -5 \).

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