f(x) = x? - 16 / x² -3x - 4 x (a) what is domain (b) where is it undefined

Calculus: Early Transcendentals
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Author:James Stewart
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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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**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 \( x = 7 \).
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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