3x + 2 Find the domain of: f(x) = x2 + x - 6 We will set x² + x - 6 equal to 0 and solve for x. We don't need to examine the numerator of the rational expression; it can be any value, including 0. The domain of the function includes all real numbers, except those that make the denominator equal to 0. From the set of real numbers, we must exclude all values of x that make the denominator 0. To find these values, we set x2 + x - 6 equal to 0 and solve for x. x2 + x - 6 = 0 Set the denominator equal to 0. (x +O)cx - <- 2) = 0 Factor the trinomial. x + 3 = 0 x - 2 = 0 Set each factor equal to 0. or x = -3 Solve each linear equation. Thus the domain of the function is the set of all real numbers except –3 and 2. Using set-builder notation we can describe the domain as {x|x is a real number and x + -3, x # 2. In interval notation, the domain is (-o, -3) U (-3, 2) U (2, 0). To check the answers, substitute -3 and 2 for x in x + x - 6 and verify that the result is 0 for each. Find the domain of f(x) = X + 2. (Enter the interval that contains smaller numbers first.) x + 8
3x + 2 Find the domain of: f(x) = x2 + x - 6 We will set x² + x - 6 equal to 0 and solve for x. We don't need to examine the numerator of the rational expression; it can be any value, including 0. The domain of the function includes all real numbers, except those that make the denominator equal to 0. From the set of real numbers, we must exclude all values of x that make the denominator 0. To find these values, we set x2 + x - 6 equal to 0 and solve for x. x2 + x - 6 = 0 Set the denominator equal to 0. (x +O)cx - <- 2) = 0 Factor the trinomial. x + 3 = 0 x - 2 = 0 Set each factor equal to 0. or x = -3 Solve each linear equation. Thus the domain of the function is the set of all real numbers except –3 and 2. Using set-builder notation we can describe the domain as {x|x is a real number and x + -3, x # 2. In interval notation, the domain is (-o, -3) U (-3, 2) U (2, 0). To check the answers, substitute -3 and 2 for x in x + x - 6 and verify that the result is 0 for each. Find the domain of f(x) = X + 2. (Enter the interval that contains smaller numbers first.) x + 8
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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![### Finding the Domain of a Rational Function
**Example Problem:**
Find the domain of: \( f(x) = \frac{3x + 2}{x^2 + x - 6} \).
**Step-by-Step Solution:**
1. **Identify the Denominator:**
We will set \(x^2 + x - 6\) to 0 and solve for \(x\).
2. **Set up the Equation:**
\[
x^2 + x - 6 = 0
\]
3. **Factor the Trinomial:**
\[
(x + 3)(x - 2) = 0
\]
4. **Solve Each Linear Equation:**
- \( x + 3 = 0 \Rightarrow x = -3 \)
- \( x - 2 = 0 \Rightarrow x = 2 \)
5. **Exclude These Values:**
From the set of real numbers, we must exclude all values of \(x\) that make the denominator zero. Thus, \(x \neq -3\) and \(x \neq 2\).
6. **Express Domain:**
- **Set-Builder Notation:** \(\{ x \mid x \text{ is a real number and } x \neq -3, x \neq 2 \}\).
- **Interval Notation:** The domain is \( (-\infty, -3) \cup (-3, 2) \cup (2, \infty) \).
7. **Verification:**
To check, substitute \(x = -3\) and \(x = 2\) into the original rational expression \(x^2 + x - 6\) and verify that the result is 0 for each.
**Additional Example:**
Find the domain of \( f(x) = \frac{x^2 + 2}{x + 8} \) (Enter the interval that contains smaller numbers first.)
- **Exclude \( x = -8 \) from the domain**, since setting the denominator \( x + 8 = 0 \) leads to \( x = -8 \).
- **Express Domain:**
- **Interval Notation:** \( (-\infty, -8) \cup (-8, \infty) \).
\(
\begin{align*}
\big](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F758f6220-2c9c-498c-a499-60dbbe8461af%2F7d71c140-ae23-450e-bd60-7f6d1dfd7b1c%2F6vd9a69_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Domain of a Rational Function
**Example Problem:**
Find the domain of: \( f(x) = \frac{3x + 2}{x^2 + x - 6} \).
**Step-by-Step Solution:**
1. **Identify the Denominator:**
We will set \(x^2 + x - 6\) to 0 and solve for \(x\).
2. **Set up the Equation:**
\[
x^2 + x - 6 = 0
\]
3. **Factor the Trinomial:**
\[
(x + 3)(x - 2) = 0
\]
4. **Solve Each Linear Equation:**
- \( x + 3 = 0 \Rightarrow x = -3 \)
- \( x - 2 = 0 \Rightarrow x = 2 \)
5. **Exclude These Values:**
From the set of real numbers, we must exclude all values of \(x\) that make the denominator zero. Thus, \(x \neq -3\) and \(x \neq 2\).
6. **Express Domain:**
- **Set-Builder Notation:** \(\{ x \mid x \text{ is a real number and } x \neq -3, x \neq 2 \}\).
- **Interval Notation:** The domain is \( (-\infty, -3) \cup (-3, 2) \cup (2, \infty) \).
7. **Verification:**
To check, substitute \(x = -3\) and \(x = 2\) into the original rational expression \(x^2 + x - 6\) and verify that the result is 0 for each.
**Additional Example:**
Find the domain of \( f(x) = \frac{x^2 + 2}{x + 8} \) (Enter the interval that contains smaller numbers first.)
- **Exclude \( x = -8 \) from the domain**, since setting the denominator \( x + 8 = 0 \) leads to \( x = -8 \).
- **Express Domain:**
- **Interval Notation:** \( (-\infty, -8) \cup (-8, \infty) \).
\(
\begin{align*}
\big
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