Find The Domain And Range Of The Function $f(x)=\left\{\begin{array}{l} 3x+4, -1\ \textless \ X\ \textless \ 2 \\ 1+x, 2 \leqslant X\ \textless \ 5 \end{array}\right.$

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. These sub-functions are often referred to as "pieces" of the function, and they are combined to form a single function. In this article, we will explore how to find the domain and range of a piecewise function, using the function $f(x)=\left\{\begin{array}{l} 3x+4, -1\ \textless \ x\ \textless \ 2 \\ 1+x, 2 \leqslant x\ \textless \ 5 \end{array}\right.$ as an example.

Understanding the Piecewise Function

Before we can find the domain and range of the function, we need to understand how it is defined. The function $f(x)$ is a piecewise function, meaning that it is defined by two sub-functions:

• For $-1 < x < 2$, the function is defined as $f(x) = 3x + 4$.
• For $2 \leq x < 5$, the function is defined as $f(x) = 1 + x$.

Finding the Domain

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ for which the function is valid. To find the domain of the piecewise function, we need to consider the intervals on which each sub-function is defined.

For the first sub-function, $f(x) = 3x + 4$, the domain is $-1 < x < 2$. This means that the function is defined for all values of $x$ between $-1$ and $2$, but not including $-1$ and $2$.

For the second sub-function, $f(x) = 1 + x$, the domain is $2 \leq x < 5$. This means that the function is defined for all values of $x$ greater than or equal to $2$, but less than $5$.

Combining the Domains

To find the complete domain of the piecewise function, we need to combine the domains of the two sub-functions. The domain of the first sub-function is $-1 < x < 2$, and the domain of the second sub-function is $2 \leq x < 5$. Since these two intervals do not overlap, we can simply combine them to form the complete domain of the piecewise function.

The complete domain of the piecewise function is $-1 < x < 5$.

Finding the Range

The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of $f(x)$ for which the function is valid. To find the range of the piecewise function, we need to consider the behavior of each sub-function on its respective interval.

For the first sub-function, $f(x) = 3x + 4$, we can find the range by considering the behavior of the function as $x$ approaches the endpoints of the interval. As $x$ approaches $-1$ from the left, $f(x)$ approaches $5$. As $x$ approaches $2$ from the right, $f(x)$ approaches $10$.

For the second sub-function, $f(x) = 1 + x$, we can find the range by considering the behavior of the function as $x$ approaches the endpoints of the interval. As $x$ approaches $2$ from the right, $f(x)$ approaches $3$. As $x$ approaches $5$ from the left, $f(x)$ approaches $6$.

Combining the Ranges

To find the complete range of the piecewise function, we need to combine the ranges of the two sub-functions. The range of the first sub-function is $5 \leq f(x) \leq 10$, and the range of the second sub-function is $3 \leq f(x) \leq 6$. Since these two intervals overlap, we need to find the intersection of the two intervals to form the complete range of the piecewise function.

The complete range of the piecewise function is $3 \leq f(x) \leq 10$.

Conclusion

In this article, we have explored how to find the domain and range of a piecewise function. We have used the function $f(x)=\left\{\begin{array}{l} 3x+4, -1\ \textless \ x\ \textless \ 2 \\ 1+x, 2 \leqslant x\ \textless \ 5 \end{array}\right.$ as an example, and have shown how to find the domain and range of the function by considering the behavior of each sub-function on its respective interval. We have also shown how to combine the domains and ranges of the two sub-functions to form the complete domain and range of the piecewise function.

Final Answer

The domain of the piecewise function is $-1 < x < 5$, and the range of the piecewise function is $3 \leq f(x) \leq 10$.

Key Takeaways

• The domain of a piecewise function is the set of all possible input values for which the function is defined.
• The range of a piecewise function is the set of all possible output values for which the function is defined.
• To find the domain and range of a piecewise function, we need to consider the behavior of each sub-function on its respective interval.
• We can combine the domains and ranges of the two sub-functions to form the complete domain and range of the piecewise function.

Further Reading

If you are interested in learning more about piecewise functions and how to find their domain and range, we recommend checking out the following resources:

• Khan Academy: Piecewise Functions
• Mathway: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

These resources provide a more in-depth explanation of piecewise functions and how to find their domain and range. They also provide examples and practice problems to help you understand the concept better.

Introduction

In our previous article, we explored how to find the domain and range of a piecewise function. We used the function $f(x)=\left\{\begin{array}{l} 3x+4, -1\ \textless \ x\ \textless \ 2 \\ 1+x, 2 \leqslant x\ \textless \ 5 \end{array}\right.$ as an example, and showed how to find the domain and range of the function by considering the behavior of each sub-function on its respective interval. In this article, we will answer some frequently asked questions about finding the domain and range of a piecewise function.

Q: What is the domain of a piecewise function?

A: The domain of a piecewise function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ for which the function is valid.

Q: How do I find the domain of a piecewise function?

A: To find the domain of a piecewise function, you need to consider the behavior of each sub-function on its respective interval. You can do this by finding the values of $x$ for which each sub-function is defined.

Q: What is the range of a piecewise function?

A: The range of a piecewise function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of $f(x)$ for which the function is valid.

Q: How do I find the range of a piecewise function?

A: To find the range of a piecewise function, you need to consider the behavior of each sub-function on its respective interval. You can do this by finding the values of $f(x)$ for which each sub-function is defined.

Q: Can I have multiple domains and ranges for a piecewise function?

A: Yes, you can have multiple domains and ranges for a piecewise function. This is because each sub-function has its own domain and range, and these can be different from one another.

Q: How do I combine the domains and ranges of multiple sub-functions?

A: To combine the domains and ranges of multiple sub-functions, you need to find the intersection of the domains and the union of the ranges. This will give you the complete domain and range of the piecewise function.

Q: What if the domains and ranges of the sub-functions overlap?

A: If the domains and ranges of the sub-functions overlap, you need to find the intersection of the domains and the union of the ranges. This will give you the complete domain and range of the piecewise function.

Q: Can I have a piecewise function with an infinite number of sub-functions?

A: Yes, you can have a piecewise function with an infinite number of sub-functions. This is because each sub-function can have its own domain and range, and these can be different from one another.

Q: How do I find the domain and range of a piecewise function with an infinite number of sub-functions?

A: To find the domain and range of a piecewise function with an infinite number of sub-functions, you need to consider the behavior of each sub-function on its respective interval. You can do this by finding the values of $x$ and $f(x)$ for which each sub-function is defined.

Q: Are there any special cases I should be aware of when finding the domain and range of a piecewise function?

A: Yes, there are several special cases you should be aware of when finding the domain and range of a piecewise function. These include:

• If a sub-function is defined for all values of $x$, then the domain of the piecewise function is the entire real line.
• If a sub-function is defined for no values of $x$, then the domain of the piecewise function is the empty set.
• If a sub-function is defined for a single value of $x$, then the domain of the piecewise function is a single point.
• If a sub-function is defined for a finite number of values of $x$, then the domain of the piecewise function is a finite set.

Conclusion

In this article, we have answered some frequently asked questions about finding the domain and range of a piecewise function. We have shown how to find the domain and range of a piecewise function, and have discussed some special cases that you should be aware of when finding the domain and range of a piecewise function.

Final Answer

The domain of a piecewise function is the set of all possible input values for which the function is defined, and the range of a piecewise function is the set of all possible output values for which the function is defined.

Key Takeaways

• The domain of a piecewise function is the set of all possible input values for which the function is defined.
• The range of a piecewise function is the set of all possible output values for which the function is defined.
• To find the domain and range of a piecewise function, you need to consider the behavior of each sub-function on its respective interval.
• You can combine the domains and ranges of multiple sub-functions to form the complete domain and range of the piecewise function.

Further Reading

If you are interested in learning more about piecewise functions and how to find their domain and range, we recommend checking out the following resources:

• Khan Academy: Piecewise Functions
• Mathway: Piecewise Functions
• Wolfram MathWorld: Piecewise Functions

These resources provide a more in-depth explanation of piecewise functions and how to find their domain and range. They also provide examples and practice problems to help you understand the concept better.