### Understanding Functions and GraphsIn mathematics, a function $ f(x) $ is a relation between a set of inputs (known as the domain) and a set of possible outputs (known as the range) where each input is related to exactly one output.The notation $ f(x) = $ is often used to describe this relation. Here, $ f(x) $ represents the output of the function corresponding to the input $ x $.#### Diagram DescriptionThe image shows the expression $ f(x) = $ followed by a blank rectangular space. This blank space typically represents a placeholder for either a graphical depiction of $ f(x) $ or the explicit form of the function itself, such as a mathematical equation.In educational settings, this format is commonly used to:1. **Illustrate Graphs**: Depict the graph of the function defined by $ f(x) $.2. **Determine Functions**: Challenge students to derive the functional form or plot points on a graph manually.3. **Explain Concepts**: Visually explain how varying $ x $ affects $ f(x) $ through a graphical representation.For a detailed example, imagine the rectangular space contains a graph displaying the behavior of a quadratic function like $ f(x) = x^2 - 4x + 3 $. This graph would typically be a parabola opening upwards, showing key points such as the vertex and intercepts.### SummaryThe notation $ f(x) = $ followed by a space or graph is a fundamental component in understanding functions, as it helps in visualizing, interpreting, and analyzing mathematical relationships. It is crucial for developing a deeper comprehension of how inputs to a function produce specific outputs, and how these can be represented and understood graphically. ### Mathematical Function Transformation#### Function RepresentationThe function presented is:\[ r(x) = -\frac{x^2}{3} + 2 \]#### TaskDetermine the more basic function that has been shifted, reflected, stretched, or compressed.#### AnalysisThis function can be broken down into transformations of the basic quadratic function $ f(x) = x^2 $:1. **Reflection**: The coefficient $-\frac{1}{3}$ represents a reflection over the x-axis.2. **Compression**: The factor $\frac{1}{3}$ indicates that the function is vertically compressed by a factor of 3.3. **Vertical Shift**: The "+2" at the end of the function indicates that the graph is shifted upward by 2 units.Thus, the transformations applied to the basic function $ f(x) = x^2 $ to obtain $ r(x) $ include a reflection over the x-axis, a vertical compression by a factor of $\frac{1}{3}$, and an upward shift by 2 units.Understanding these transformations is crucial for analyzing the behavior and shape of the given function's graph.

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Math

Calculus

flx)

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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Question

### Understanding Functions and Graphs

In mathematics, a function $ f(x) $ is a relation between a set of inputs (known as the domain) and a set of possible outputs (known as the range) where each input is related to exactly one output.

The notation $ f(x) = $ is often used to describe this relation. Here, $ f(x) $ represents the output of the function corresponding to the input $ x $.

#### Diagram Description
The image shows the expression $ f(x) = $ followed by a blank rectangular space. This blank space typically represents a placeholder for either a graphical depiction of $ f(x) $ or the explicit form of the function itself, such as a mathematical equation.

In educational settings, this format is commonly used to:
1. **Illustrate Graphs**: Depict the graph of the function defined by $ f(x) $.
2. **Determine Functions**: Challenge students to derive the functional form or plot points on a graph manually.
3. **Explain Concepts**: Visually explain how varying $ x $ affects $ f(x) $ through a graphical representation.

For a detailed example, imagine the rectangular space contains a graph displaying the behavior of a quadratic function like $ f(x) = x^2 - 4x + 3 $. This graph would typically be a parabola opening upwards, showing key points such as the vertex and intercepts.

### Summary
The notation $ f(x) = $ followed by a space or graph is a fundamental component in understanding functions, as it helps in visualizing, interpreting, and analyzing mathematical relationships. It is crucial for developing a deeper comprehension of how inputs to a function produce specific outputs, and how these can be represented and understood graphically.

### Mathematical Function Transformation

#### Function Representation

The function presented is:

\[ r(x) = -\frac{x^2}{3} + 2 \]

#### Task

Determine the more basic function that has been shifted, reflected, stretched, or compressed.

#### Analysis

This function can be broken down into transformations of the basic quadratic function $ f(x) = x^2 $:

1. **Reflection**: The coefficient $-\frac{1}{3}$ represents a reflection over the x-axis.
2. **Compression**: The factor $\frac{1}{3}$ indicates that the function is vertically compressed by a factor of 3.
3. **Vertical Shift**: The "+2" at the end of the function indicates that the graph is shifted upward by 2 units.

Thus, the transformations applied to the basic function $ f(x) = x^2 $ to obtain $ r(x) $ include a reflection over the x-axis, a vertical compression by a factor of $\frac{1}{3}$, and an upward shift by 2 units.

Understanding these transformations is crucial for analyzing the behavior and shape of the given function's graph.

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