Use The Drawing Tools To Form The Correct Answers On The Graph.Consider Function { F $} . . . { f(x)=\begin{cases} \left(\frac{1}{2}\right)^x, & X \leq 0 \\ 2^x, & X \ \textgreater \ 0 \end{cases} \} Complete The Table Of Values

Introduction

In mathematics, 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 the graph of a piecewise function, specifically the function $f(x)$ defined as:

$$f(x)=\begin{cases} \left(\frac{1}{2}\right)^x, & x \leq 0 \\ 2^x, & x \ \textgreater \ 0 \end{cases}$$

Understanding the Piecewise Function

The piecewise function $f(x)$ is defined as two separate functions, one for $x \leq 0$ and another for $x > 0$. The function for $x \leq 0$ is $\left(\frac{1}{2}\right)^x$, which is an exponential function with base $\frac{1}{2}$. The function for $x > 0$ is $2^x$, which is also an exponential function with base $2$.

Graphing the Piecewise Function

To graph the piecewise function, we need to graph the two separate functions and then combine them into a single graph. For the function $\left(\frac{1}{2}\right)^x$, we can graph it by plotting points for $x = -1, -2, -3, ...$ and connecting them with a smooth curve. For the function $2^x$, we can graph it by plotting points for $x = 1, 2, 3, ...$ and connecting them with a smooth curve.

Completing the Table of Values

To complete the table of values, we need to find the values of $f(x)$ for $x = -2, -1, 0, 1, 2, 3, ...$. We can do this by plugging in the values of $x$ into the piecewise function and evaluating the expression.

Analyzing the Graph

The graph of the piecewise function $f(x)$ consists of two separate branches, one for $x \leq 0$ and another for $x > 0$. The branch for $x \leq 0$ is a decreasing exponential function, while the branch for $x > 0$ is an increasing exponential function. The two branches meet at the point $(0, 1)$.

Key Features of the Graph

• The graph has two separate branches, one for $x \leq 0$ and another for $x > 0$.
• The branch for $x \leq 0$ is a decreasing exponential function.
• The branch for $x > 0$ is an increasing exponential function.
• The two branches meet at the point $(0, 1)$.

Conclusion

In this article, we explored the graph of a piecewise function, specifically the function $f(x)$ defined as:

$$f(x)=\begin{cases} \left(\frac{1}{2}\right)^x, & x \leq 0 \\ 2^x, & x \ \textgreater \ 0 \end{cases}$$

We completed the table of values and analyzed the graph, identifying key features such as the two separate branches and the meeting point at $(0, 1)$. This article provides a comprehensive understanding of the graph of a piecewise function and its key features.

Using the Drawing Tools to Form the Correct Answers on the Graph

To form the correct answers on the graph, we can use the following steps:

1. Draw the branch for $x \leq 0$: Draw a decreasing exponential function for $x \leq 0$.
2. Draw the branch for $x > 0$: Draw an increasing exponential function for $x > 0$.
3. Meet at the point $(0, 1)$: Ensure that the two branches meet at the point $(0, 1)$.

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph the two separate functions and then combine them into a single graph. For the function $\left(\frac{1}{2}\right)^x$, you can graph it by plotting points for $x = -1, -2, -3, ...$ and connecting them with a smooth curve. For the function $2^x$, you can graph it by plotting points for $x = 1, 2, 3, ...$ and connecting them with a smooth curve.

Q: What are the key features of the graph of a piecewise function?

A: The key features of the graph of a piecewise function include:

• Two separate branches, one for $x \leq 0$ and another for $x > 0$
• The branch for $x \leq 0$ is a decreasing exponential function
• The branch for $x > 0$ is an increasing exponential function
• The two branches meet at the point $(0, 1)$

Q: How do I complete the table of values for a piecewise function?

A: To complete the table of values for a piecewise function, you need to find the values of $f(x)$ for $x = -2, -1, 0, 1, 2, 3, ...$. You can do this by plugging in the values of $x$ into the piecewise function and evaluating the expression.

Q: What is the meeting point of the two branches of a piecewise function?

A: The meeting point of the two branches of a piecewise function is the point where the two branches intersect. In the case of the function $f(x)$, the meeting point is $(0, 1)$.

Q: How do I use the drawing tools to form the correct answers on the graph of a piecewise function?

A: To use the drawing tools to form the correct answers on the graph of a piecewise function, you can follow these steps:

1. Draw the branch for $x \leq 0$: Draw a decreasing exponential function for $x \leq 0$.
2. Draw the branch for $x > 0$: Draw an increasing exponential function for $x > 0$.
3. Meet at the point $(0, 1)$: Ensure that the two branches meet at the point $(0, 1)$.

Q: What are some common mistakes to avoid when graphing a piecewise function?

A: Some common mistakes to avoid when graphing a piecewise function include:

• Not drawing the two separate branches
• Not meeting at the correct point
• Not using the correct function for each branch

Q: How do I check my work when graphing a piecewise function?

A: To check your work when graphing a piecewise function, you can:

• Verify that the two branches meet at the correct point
• Check that the functions are correctly applied to each branch
• Use a table of values to check that the function is correctly evaluated for each value of $x$

By following these steps and avoiding common mistakes, you can ensure that your graph of a piecewise function is accurate and complete.