Identify four ordered pairs on the graph of the function. ### Understanding Rational FunctionsThe function displayed is:\[ f(x) = \frac{x^2 + 10}{x^3 - 8} \]### Exploring the FunctionThis is a rational function where:- The **numerator** is $x^2 + 10$.- The **denominator** is $x^3 - 8$.### Key Elements of the Function1. **Domain**: The function is undefined where the denominator is zero. Solve $x^3 - 8 = 0$ to find these values.2. **Asymptotes**: - **Vertical Asymptotes**: Occur where the denominator is zero, provided the numerator does not also equal zero at these points. - **Horizontal Asymptotes**: Determined by the degrees of the polynomials in the numerator and denominator.3. **Intercepts**: - **Y-intercept**: Set $x = 0$ and solve for $f(x)$. - **X-intercepts**: Set the numerator equal to zero and solve for $x$ (if possible).### Graphing the FunctionTo understand the behavior of the graph:- Identify any **asymptotes** and **intercepts**.- Analyze the **end behavior** by considering the degrees of the numerator and denominator.This information helps create a comprehensive graph of the function.

Answered: x² + 10 f(x) = %3D x3 - 8 graph of the…

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x² + 10 f(x) = %3D x3 - 8 graph of the function.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

Identify four ordered pairs on the graph of the function.

### Understanding Rational Functions

The function displayed is:

\[ f(x) = \frac{x^2 + 10}{x^3 - 8} \]

### Exploring the Function

This is a rational function where:

- The **numerator** is $x^2 + 10$.
- The **denominator** is $x^3 - 8$.

### Key Elements of the Function

1. **Domain**: The function is undefined where the denominator is zero. Solve $x^3 - 8 = 0$ to find these values.

2. **Asymptotes**:
- **Vertical Asymptotes**: Occur where the denominator is zero, provided the numerator does not also equal zero at these points.
- **Horizontal Asymptotes**: Determined by the degrees of the polynomials in the numerator and denominator.

3. **Intercepts**:
- **Y-intercept**: Set $x = 0$ and solve for $f(x)$.
- **X-intercepts**: Set the numerator equal to zero and solve for $x$ (if possible).

### Graphing the Function

To understand the behavior of the graph:

- Identify any **asymptotes** and **intercepts**.
- Analyze the **end behavior** by considering the degrees of the numerator and denominator.

This information helps create a comprehensive graph of the function.

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