x+5 What is the domain of ƒ(x)= x² - 6x+8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Text:**
What is the domain of \( f(x) = \frac{x+5}{x^2 - 6x + 8} \)?
**Explanation:**
To determine the domain of the function \( f(x) = \frac{x+5}{x^2 - 6x + 8} \), we need to consider the values of \( x \) for which the function is defined. This function is a rational function, and it is undefined when the denominator is zero.
To find these values, set the denominator equal to zero and solve for \( x \):
\[ x^2 - 6x + 8 = 0 \]
Factor the quadratic expression:
\[ (x - 2)(x - 4) = 0 \]
The solutions to this equation are \( x = 2 \) and \( x = 4 \). These are the values that make the denominator zero, so they need to be excluded from the domain.
**Domain:**
The domain of \( f(x) \) is all real numbers except \( x = 2 \) and \( x = 4 \). In interval notation, the domain is:
\( (-\infty, 2) \cup (2, 4) \cup (4, \infty) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc84d0291-1107-4711-af5a-30afb213fbbf%2Ffac8a83b-fa52-4450-8fff-7a06e1303418%2Fw4bc6z_processed.png&w=3840&q=75)
Transcribed Image Text:**Text:**
What is the domain of \( f(x) = \frac{x+5}{x^2 - 6x + 8} \)?
**Explanation:**
To determine the domain of the function \( f(x) = \frac{x+5}{x^2 - 6x + 8} \), we need to consider the values of \( x \) for which the function is defined. This function is a rational function, and it is undefined when the denominator is zero.
To find these values, set the denominator equal to zero and solve for \( x \):
\[ x^2 - 6x + 8 = 0 \]
Factor the quadratic expression:
\[ (x - 2)(x - 4) = 0 \]
The solutions to this equation are \( x = 2 \) and \( x = 4 \). These are the values that make the denominator zero, so they need to be excluded from the domain.
**Domain:**
The domain of \( f(x) \) is all real numbers except \( x = 2 \) and \( x = 4 \). In interval notation, the domain is:
\( (-\infty, 2) \cup (2, 4) \cup (4, \infty) \)
Expert Solution
Step 1: Definition
For defining domain of rational polynomial denumerator should not be zero.
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