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...
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Transcribed Image Text:**Mathematics: Understanding Rational Functions**
**Function Analysis**
• \( f(x) = \frac{x^2 - 16}{x^2 - 3x - 4x} \)
**Questions to Explore:**
(a) What is the domain of the function?
(b) Where is the function undefined?
**Explanation:**
- **Domain**: The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- **Undefined Points**: A rational function is undefined where its denominator equals zero.
**Detailed Steps:**
1. **Identify the Denominator**:
The function \( f(x) \) is given as \( \frac{x^2 - 16}{x^2 - 3x - 4x} \). To find where the function is undefined, we need to set the denominator equal to zero and solve for x.
2. **Solve Denominator Equals Zero**:
The denominator \( x^2 - 3x - 4x \) simplifies to \( x^2 - 7x \). Setting it to zero:
\( x^2 - 7x = 0 \)
Factoring out x:
\( x(x - 7) = 0 \)
Thus, \( x = 0 \) or \( x = 7 \).
3. **Domain**:
To determine the domain, we exclude the points where the function is undefined. Therefore, the domain of \( f(x) \) is:
\( \{x \in \mathbb{R} \mid x \neq 0, x \neq 7\} \)
4. **Undefined Points**:
The function \( f(x) \) is undefined at \( x = 0 \) and \( x = 7 \).
In conclusion:
- **Domain**: All real numbers except \( x = 0 \) and \( x = 7 \).
- **Undefined Points**: The function \( f(x) \) is undefined at \( x = 0 \) and \( x = 7 \).
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