Which Graph Represents The Function?${ f(x)= \begin{cases} (x+2)^2 & \text{if } X \ \textless \ -1 \ |x| - 1 & \text{if } -1 \leq X \leq 1 \ -\sqrt[3]{x} & \text{if } X \ \textgreater \ 1 \end{cases} }$

Introduction

When dealing with piecewise functions, it can be challenging to determine which graph represents the function. A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In this article, we will explore how to determine which graph represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

Understanding the Function

To determine which graph represents the function, we need to understand the behavior of each sub-function. Let's analyze each sub-function separately:

Sub-function 1: $(x+2)^2$

This sub-function is defined for $x < -1$. It represents a quadratic function that is shifted 2 units to the left. The graph of this sub-function is a parabola that opens upwards, with its vertex at $x = -2$.

Sub-function 2: $|x| - 1$

This sub-function is defined for $-1 \leq x \leq 1$. It represents the absolute value function shifted 1 unit downwards. The graph of this sub-function is a V-shaped graph with its vertex at $x = 0$.

Sub-function 3: $-\sqrt[3]{x}$

This sub-function is defined for $x > 1$. It represents a cubic root function that is reflected across the x-axis. The graph of this sub-function is a curve that decreases as $x$ increases.

Analyzing the Graphs

Now that we have a good understanding of each sub-function, let's analyze the graphs that represent the function. We will consider three possible graphs:

Graph 1

This graph represents a parabola that opens upwards, with its vertex at $x = -2$. It is defined for $x < -1$.

Graph 2

This graph represents a V-shaped graph with its vertex at $x = 0$. It is defined for $-1 \leq x \leq 1$.

Graph 3

This graph represents a curve that decreases as $x$ increases. It is defined for $x > 1$.

Determining the Correct Graph

To determine which graph represents the function, we need to consider the behavior of each sub-function. Let's analyze each graph separately:

Graph 1

This graph represents a parabola that opens upwards, with its vertex at $x = -2$. However, this sub-function is only defined for $x < -1$, which means that the graph should not be visible for $x \geq -1$. Therefore, this graph is not the correct representation of the function.

Graph 2

This graph represents a V-shaped graph with its vertex at $x = 0$. However, this sub-function is only defined for $-1 \leq x \leq 1$, which means that the graph should not be visible for $x < -1$ and $x > 1$. Therefore, this graph is not the correct representation of the function.

Graph 3

This graph represents a curve that decreases as $x$ increases. However, this sub-function is only defined for $x > 1$, which means that the graph should not be visible for $x \leq 1$. Therefore, this graph is not the correct representation of the function.

Conclusion

After analyzing each graph separately, we can conclude that none of the graphs represent the function correctly. However, we can combine the sub-functions to create a single graph that represents the function.

The Correct Graph

The correct graph represents a piecewise function that combines the sub-functions. It is defined as follows:

• For $x < -1$, the graph represents a parabola that opens upwards, with its vertex at $x = -2$.
• For $-1 \leq x \leq 1$, the graph represents a V-shaped graph with its vertex at $x = 0$.
• For $x > 1$, the graph represents a curve that decreases as $x$ increases.

The final graph represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

Final Answer

The final answer is the graph that represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

This graph represents a piecewise function that combines the sub-functions. It is defined as follows:

• For $x < -1$, the graph represents a parabola that opens upwards, with its vertex at $x = -2$.
• For $-1 \leq x \leq 1$, the graph represents a V-shaped graph with its vertex at $x = 0$.
• For $x > 1$, the graph represents a curve that decreases as $x$ increases.

The final graph represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

Introduction

In our previous article, we explored how to determine which graph represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

We analyzed each sub-function and determined that the correct graph represents a piecewise function that combines the sub-functions. In this article, we will answer some frequently asked questions about the graph and the function.

Q&A

Q: What is the domain of the function?

A: The domain of the function is all real numbers, $x \in \mathbb{R}$.

Q: What is the range of the function?

A: The range of the function is all real numbers, $y \in \mathbb{R}$.

Q: What is the behavior of the function for $x < -1$?

A: For $x < -1$, the function represents a parabola that opens upwards, with its vertex at $x = -2$.

Q: What is the behavior of the function for $-1 \leq x \leq 1$?

A: For $-1 \leq x \leq 1$, the function represents a V-shaped graph with its vertex at $x = 0$.

Q: What is the behavior of the function for $x > 1$?

A: For $x > 1$, the function represents a curve that decreases as $x$ increases.

Q: Is the function continuous?

A: The function is continuous at $x = -1$ and $x = 1$, but it is not continuous at $x = 1$.

Q: Is the function differentiable?

A: The function is differentiable at $x = -1$ and $x = 1$, but it is not differentiable at $x = 1$.

Q: Can the function be simplified?

A: The function cannot be simplified, as it is a piecewise function.

Q: Can the function be graphed using a single equation?

A: No, the function cannot be graphed using a single equation, as it is a piecewise function.

Q: Can the function be evaluated at a specific point?

A: Yes, the function can be evaluated at a specific point by substituting the value of $x$ into the function.

Conclusion

In this article, we answered some frequently asked questions about the graph and the function. We hope that this article has been helpful in understanding the behavior of the function and the graph that represents it.

Final Answer

The final answer is the graph that represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}

This graph represents a piecewise function that combines the sub-functions. It is defined as follows:

• For $x < -1$, the graph represents a parabola that opens upwards, with its vertex at $x = -2$.
• For $-1 \leq x \leq 1$, the graph represents a V-shaped graph with its vertex at $x = 0$.
• For $x > 1$, the graph represents a curve that decreases as $x$ increases.

The final graph represents the function $f(x)$, given by:

{ f(x)= \begin{cases} (x+2)^2 & \text{if } x \ \textless \ -1 \\ |x| - 1 & \text{if } -1 \leq x \leq 1 \\ -\sqrt[3]{x} & \text{if } x \ \textgreater \ 1 \end{cases} \}