Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**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:
\( (-\infty, 2) \cup (2, 4) \cup (4, \infty) \)](https://content.bartleby.com/qna-images/question/c84d0291-1107-4711-af5a-30afb213fbbf/fac8a83b-fa52-4450-8fff-7a06e1303418/w4bc6z_processed.png)
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
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Step 1: Definition
For defining domain of rational polynomial denumerator should not be zero.
Step by stepSolved in 3 steps with 2 images
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