Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Find the domain of the given function (and explain how to get the answer).
![Title: Understanding the Domain of a Rational Function
---
**Objective:** Determine the domain of the given rational function.
**Function:**
\[
f(x) = \frac{5^x}{x^2 - 5x + 6}
\]
**Explanation:**
1. **Numerator:** \( 5^x \) is an exponential function defined for all real numbers. It does not affect the domain.
2. **Denominator:** \( x^2 - 5x + 6 \) is a quadratic expression. The domain is affected by the values of \( x \) that make the denominator zero, as division by zero is undefined.
3. **Domain Determination:**
- Factor the quadratic in the denominator: \( x^2 - 5x + 6 = (x - 2)(x - 3) \).
- Set the factors equal to zero to find the values that are not in the domain:
- \( x - 2 = 0 \) gives \( x = 2 \)
- \( x - 3 = 0 \) gives \( x = 3 \)
4. **Conclusion:**
- The domain of \( f(x) \) is all real numbers except \( x = 2 \) and \( x = 3 \).
This function is defined for all real numbers except where the denominator equals zero. Hence, the values \( x = 2 \) and \( x = 3 \) are excluded from the domain, which is the set of all real numbers except for these two values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcea6d79-c0c2-49cd-aa31-c3eca199518f%2F0126296b-6b24-4253-a8ab-a52f7ef0ee2b%2Fn6t75ul_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Understanding the Domain of a Rational Function
---
**Objective:** Determine the domain of the given rational function.
**Function:**
\[
f(x) = \frac{5^x}{x^2 - 5x + 6}
\]
**Explanation:**
1. **Numerator:** \( 5^x \) is an exponential function defined for all real numbers. It does not affect the domain.
2. **Denominator:** \( x^2 - 5x + 6 \) is a quadratic expression. The domain is affected by the values of \( x \) that make the denominator zero, as division by zero is undefined.
3. **Domain Determination:**
- Factor the quadratic in the denominator: \( x^2 - 5x + 6 = (x - 2)(x - 3) \).
- Set the factors equal to zero to find the values that are not in the domain:
- \( x - 2 = 0 \) gives \( x = 2 \)
- \( x - 3 = 0 \) gives \( x = 3 \)
4. **Conclusion:**
- The domain of \( f(x) \) is all real numbers except \( x = 2 \) and \( x = 3 \).
This function is defined for all real numbers except where the denominator equals zero. Hence, the values \( x = 2 \) and \( x = 3 \) are excluded from the domain, which is the set of all real numbers except for these two values.
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