Which Graph Represents The Piecewise-defined Function ${ f(x) = \left{ \begin{array}{ll} -x + 4, & 0 \leq X \ \textless \ 3 \ 6, & X \geq 3 \end{array} \right. }$

Introduction

In mathematics, a piecewise-defined 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 overall function. In this article, we will explore the concept of piecewise-defined functions and how to represent them graphically.

What is a Piecewise-Defined Function?

A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function is typically represented using a set of rules or equations, where each rule applies to a specific interval of the domain. For example, consider the function:

$$f(x) = \left\{ \begin{array}{ll} -x + 4, & 0 \leq x \ \textless \ 3 \\ 6, & x \geq 3 \end{array} \right.$$

This function is defined by two sub-functions: $-x + 4$ for $0 \leq x \ \textless \ 3$, and $6$ for $x \geq 3$. The function is piecewise-defined because it is defined by multiple sub-functions, each applied to a specific interval of the domain.

Graphical Representation of Piecewise-Defined Functions

To represent a piecewise-defined function graphically, we need to graph each sub-function separately and then combine them to form the overall graph. Let's consider the function:

$$f(x) = \left\{ \begin{array}{ll} -x + 4, & 0 \leq x \ \textless \ 3 \\ 6, & x \geq 3 \end{array} \right.$$

Graphing the First Sub-Function

The first sub-function is $-x + 4$ for $0 \leq x \ \textless \ 3$. To graph this function, we need to find the x-intercept and the y-intercept. The x-intercept is the point where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis.

To find the x-intercept, we set $f(x) = 0$ and solve for $x$:

$$-x + 4 = 0$$

$$x = 4$$

However, since the function is only defined for $0 \leq x \ \textless \ 3$, the x-intercept is not included in the graph.

To find the y-intercept, we set $x = 0$ and solve for $f(x)$:

$$f(0) = -0 + 4 = 4$$

So, the y-intercept is $(0, 4)$.

Graphing the Second Sub-Function

The second sub-function is $6$ for $x \geq 3$. To graph this function, we need to find the x-intercept and the y-intercept.

To find the x-intercept, we set $f(x) = 0$ and solve for $x$:

$$6 = 0$$

However, since the function is only defined for $x \geq 3$, there is no x-intercept.

To find the y-intercept, we set $x = 3$ and solve for $f(x)$:

$$f(3) = 6$$

So, the y-intercept is $(3, 6)$.

Combining the Sub-Functions

Now that we have graphed each sub-function separately, we can combine them to form the overall graph. The overall graph consists of two line segments: one from $(0, 4)$ to $(3, 1)$, and another from $(3, 1)$ to $(3, 6)$.

Which Graph Represents the Piecewise-Defined Function?

Based on the graphical representation of the piecewise-defined function, we can conclude that the graph that represents the function is:

Graph 1

This graph consists of two line segments: one from $(0, 4)$ to $(3, 1)$, and another from $(3, 1)$ to $(3, 6)$.

Conclusion

In conclusion, piecewise-defined functions are functions that are defined by multiple sub-functions, each applied to a specific interval of the domain. To represent a piecewise-defined function graphically, we need to graph each sub-function separately and then combine them to form the overall graph. The graph that represents the piecewise-defined function is the one that consists of two line segments: one from $(0, 4)$ to $(3, 1)$, and another from $(3, 1)$ to $(3, 6)$.

References

• [1] "Piecewise-Defined Functions" by Math Open Reference
• [2] "Graphing Piecewise-Defined Functions" by Khan Academy

Further Reading

• "Piecewise-Defined Functions" by Wolfram MathWorld
• "Graphing Piecewise-Defined Functions" by Purplemath

Mathematics Category

• [1] "Algebra" by Khan Academy
• [2] "Calculus" by MIT OpenCourseWare
• [3] "Geometry" by Math Open Reference
  Piecewise-Defined Functions: A Q&A Guide =============================================

Introduction

In our previous article, we explored the concept of piecewise-defined functions and how to represent them graphically. In this article, we will answer some frequently asked questions about piecewise-defined functions.

Q: What is a piecewise-defined function?

A: A piecewise-defined 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-defined function?

A: To graph a piecewise-defined function, you need to graph each sub-function separately and then combine them to form the overall graph.

Q: What are the different types of piecewise-defined functions?

A: There are two main types of piecewise-defined functions:

1. Step functions: These functions have a constant value for a specific interval of the domain.
2. Piecewise-linear functions: These functions have a linear value for a specific interval of the domain.

Q: How do I determine the intervals for a piecewise-defined function?

A: To determine the intervals for a piecewise-defined function, you need to identify the points where the function changes from one sub-function to another.

Q: Can I have multiple sub-functions for a piecewise-defined function?

A: Yes, you can have multiple sub-functions for a piecewise-defined function. Each sub-function is applied to a specific interval of the domain.

Q: How do I evaluate a piecewise-defined function at a specific point?

A: To evaluate a piecewise-defined function at a specific point, you need to determine which sub-function is applied to that point and then evaluate the function using that sub-function.

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

A: Yes, you can have a piecewise-defined function with an infinite number of sub-functions. However, this is not a common occurrence in mathematics.

Q: How do I use piecewise-defined functions in real-world applications?

A: Piecewise-defined functions are used in a variety of real-world applications, including:

1. Physics: Piecewise-defined functions are used to model the motion of objects under different conditions.
2. Engineering: Piecewise-defined functions are used to model the behavior of complex systems.
3. Economics: Piecewise-defined functions are used to model the behavior of economic systems.

Q: Can I use piecewise-defined functions in calculus?

A: Yes, you can use piecewise-defined functions in calculus. Piecewise-defined functions are used to model the behavior of functions in different intervals.

Conclusion

In conclusion, piecewise-defined functions are a powerful tool in mathematics that can be used to model a wide range of real-world phenomena. By understanding how to graph and evaluate piecewise-defined functions, you can apply them to a variety of problems in physics, engineering, economics, and other fields.

References

• [1] "Piecewise-Defined Functions" by Math Open Reference
• [2] "Graphing Piecewise-Defined Functions" by Khan Academy
• [3] "Piecewise-Defined Functions in Calculus" by MIT OpenCourseWare

Further Reading

• "Piecewise-Defined Functions" by Wolfram MathWorld
• "Graphing Piecewise-Defined Functions" by Purplemath
• "Piecewise-Defined Functions in Physics" by Physics Classroom

Mathematics Category

• [1] "Algebra" by Khan Academy
• [2] "Calculus" by MIT OpenCourseWare
• [3] "Geometry" by Math Open Reference