### Problem (c)**Task**: Find the domain and range of $ f $. Write each answer as an interval or union of intervals.- **Domain:** [Input box for answer]- **Range:** [Input box for answer]**Instructions**: Specify the domain and range of the function $ f $ using interval notation. Explain each element as either an interval or a union of intervals. ### Understanding Rational Functions Through GraphsThis graph represents a rational function, which is a function of the form $ f(x) = \frac{p(x)}{q(x)} $, where $ p(x) $ and $ q(x) $ are polynomials. #### Key Features of the Graph:1. **Axes and Orientation:** - The horizontal axis is labeled as $ x $. - The vertical axis is labeled as $ y $. - There is a heavy dashed line on the vertical (y-axis) at $ x = 1 $, indicating a vertical asymptote. - There is another heavy dashed line on the horizontal (x-axis) at $ y = 1 $, suggesting a horizontal asymptote.2. **Curve Behavior:** - **Left Side of the Vertical Asymptote ($ x = 1 $):** - As $ x $ approaches 1 from the left, the curve moves upwards towards infinity. - For negative values of $ x $, the curve remains below the horizontal asymptote ($ y = 1 $) and asymptotically approaches it. - **Right Side of the Vertical Asymptote ($ x = 1 $):** - As $ x $ approaches 1 from the right, the curve moves downwards towards negative infinity. - For values of $ x $ greater than 1, the curve lies above the horizontal asymptote, approaching it as $ x $ increases.3. **Asymptotes:** - **Vertical Asymptote at $ x = 1 $:** The function approaches but never touches or crosses this line. It splits the graph into two distinct sections. - **Horizontal Asymptote at $ y = 1 $:** The curve gets closer to this line as it extends further to the left and right on the x-axis. This line represents the end-behavior of the function as $ x $ approaches positive or negative infinity.#### Mathematical Insights:Understanding the asymptotic behavior and graphical features of rational functions allows for predictions about behavior in real-world applications such as physics, engineering, and economics. Vertical asymptotes often indicate points of discontinuity or significant change, while horizontal asymptotes can indicate stability or saturation levels.

Name: ### Problem (c)**Task**: Find the domain and range of $ f $. Write
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Description: ### Problem (c)**Task**: Find the domain and range of $ f $. Write each answer as an interval or union of intervals.- **Domain:** [Input box for answer]- **Range:** [Input box for answer]**Instructions**: Specify the domain and range of the functi

DOMAIN: The values of x will define the domain of a function. RANGE: The values of y or f(x) will…

Answered: 8- 6- +2+ -4- +8- to

Math

Advanced Math

8- 6- +2+ -4- +8- to

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

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Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

Topic Video

Question

### Problem (c)

**Task**: Find the domain and range of $ f $. Write each answer as an interval or union of intervals.

- **Domain:** [Input box for answer]
- **Range:** [Input box for answer]

**Instructions**: Specify the domain and range of the function $ f $ using interval notation. Explain each element as either an interval or a union of intervals.

### Understanding Rational Functions Through Graphs

This graph represents a rational function, which is a function of the form $ f(x) = \frac{p(x)}{q(x)} $, where $ p(x) $ and $ q(x) $ are polynomials.

#### Key Features of the Graph:

1. **Axes and Orientation:**
- The horizontal axis is labeled as $ x $.
- The vertical axis is labeled as $ y $.
- There is a heavy dashed line on the vertical (y-axis) at $ x = 1 $, indicating a vertical asymptote.
- There is another heavy dashed line on the horizontal (x-axis) at $ y = 1 $, suggesting a horizontal asymptote.

2. **Curve Behavior:**
- **Left Side of the Vertical Asymptote ($ x = 1 $):**
- As $ x $ approaches 1 from the left, the curve moves upwards towards infinity.
- For negative values of $ x $, the curve remains below the horizontal asymptote ($ y = 1 $) and asymptotically approaches it.
- **Right Side of the Vertical Asymptote ($ x = 1 $):**
- As $ x $ approaches 1 from the right, the curve moves downwards towards negative infinity.
- For values of $ x $ greater than 1, the curve lies above the horizontal asymptote, approaching it as $ x $ increases.

3. **Asymptotes:**
- **Vertical Asymptote at $ x = 1 $:** The function approaches but never touches or crosses this line. It splits the graph into two distinct sections.
- **Horizontal Asymptote at $ y = 1 $:** The curve gets closer to this line as it extends further to the left and right on the x-axis. This line represents the end-behavior of the function as $ x $ approaches positive or negative infinity.

#### Mathematical Insights:
Understanding the asymptotic behavior and graphical features of rational functions allows for predictions about behavior in real-world applications such as physics, engineering, and economics. Vertical asymptotes often indicate points of discontinuity or significant change, while horizontal asymptotes can indicate stability or saturation levels.

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