Determining the Correct Graph

Oct 13, 2025 by magicalceylontours.com 22 views

Introduction

In mathematics, graphing functions is a crucial concept that helps us visualize and understand the behavior of functions. When given a function, we need to determine which graph represents that function. In this article, we will explore a piecewise function and determine which graph represents it.

The Piecewise Function

The given function is a piecewise function, which means it is defined by different formulas for different intervals of the input variable x. The function is defined as:

f(x) = \left\{ \begin{array}{ll} 1 & \text{if } -1 \ \textless \ x \ \textless \ 1 \\ 3x & \text{if } 1 \leq x \leq 2 \end{array} \right.

This function has two parts: one for the interval -1 < x < 1, where f(x) = 1, and another for the interval 1 ≤ x ≤ 2, where f(x) = 3x.

Graphing the Function

To graph the function, we need to graph each part separately and then combine them. For the first part, we have a constant function f(x) = 1, which is a horizontal line at y = 1. For the second part, we have a linear function f(x) = 3x, which is a line with a slope of 3 and a y-intercept of 0.

Graph A

Graph A is a horizontal line at y = 1 for the interval -1 < x < 1, and a line with a slope of 3 and a y-intercept of 0 for the interval 1 ≤ x ≤ 2.

Graph B

Graph B is a horizontal line at y = 1 for the interval -1 < x < 1, and a line with a slope of 3 and a y-intercept of 0 for the interval 1 ≤ x ≤ 2, but it is shifted up by 1 unit.

Graph C

Graph C is a horizontal line at y = 1 for the interval -1 < x < 1, and a line with a slope of 3 and a y-intercept of 0 for the interval 1 ≤ x ≤ 2, but it is shifted down by 1 unit.

Graph D

Graph D is a horizontal line at y = 1 for the interval -1 < x < 1, and a line with a slope of 3 and a y-intercept of 0 for the interval 1 ≤ x ≤ 2, but it is shifted up by 2 units.

Determining the Correct Graph

To determine which graph represents the function, we need to compare the graph with the function definition. The function is defined as f(x) = 1 for -1 < x < 1, and f(x) = 3x for 1 ≤ x ≤ 2. Graph A represents this function perfectly, as it has a horizontal line at y = 1 for the interval -1 < x < 1, and a line with a slope of 3 and a y-intercept of 0 for the interval 1 ≤ x ≤ 2.

Conclusion

In conclusion, the correct graph that represents the function f(x) = \left{ \begin{array}{ll} 1 & \text{if } -1 \ \textless \ x \ \textless \ 1 \ 3x & \text{if } 1 \leq x \leq 2 \end{array} \right. is Graph A.

Final Answer

The final answer is Graph A.

Introduction

In the previous article, we explored a piecewise function and determined which graph represents it. In this article, we will answer some frequently asked questions (FAQs) about the piecewise function.

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by different formulas for different intervals of the input variable x. It is a function that has multiple parts, each defined by a different formula.

Q: How do I graph a piecewise function?

A: To graph a piecewise function, you need to graph each part separately and then combine them. For each part, you need to identify the formula, the interval, and the graph of the function.

Q: What is the difference between a piecewise function and a continuous function?

A: A continuous function is a function that can be graphed without any breaks or gaps. A piecewise function, on the other hand, is a function that is defined by different formulas for different intervals of the input variable x. It may have breaks or gaps in the graph.

Q: Can a piecewise function be continuous?

A: Yes, a piecewise function can be continuous. If the different parts of the function are connected at the endpoints of the intervals, then the function is continuous.

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

A: To determine the domain of a piecewise function, you need to identify the intervals for which each part of the function is defined. The domain is the set of all input values for which the function is defined.

Q: Can a piecewise function have multiple domains?

A: Yes, a piecewise function can have multiple domains. Each part of the function may have its own domain, and the domains may overlap or be disjoint.

Q: How do I evaluate a piecewise function at a given point?

A: To evaluate a piecewise function at a given point, you need to identify the part of the function that is defined at that point. You then use the formula for that part to evaluate the function at the given point.

Q: Can a piecewise function be used to model real-world phenomena?

A: Yes, a piecewise function can be used to model real-world phenomena. Piecewise functions can be used to model functions that have different behaviors in different intervals, such as the cost of a product that changes depending on the quantity ordered.

Q: How do I use a piecewise function in a real-world application?

A: To use a piecewise function in a real-world application, you need to identify the different parts of the function and the intervals for which each part is defined. You then use the function to model the real-world phenomenon and make predictions or decisions based on the function.

Conclusion

In conclusion, piecewise functions are an important concept in mathematics that can be used to model real-world phenomena. By understanding how to graph, evaluate, and use piecewise functions, you can apply them to a wide range of problems and applications.

Final Answer

The final answer is that piecewise functions are a powerful tool for modeling and analyzing complex phenomena. By understanding how to work with piecewise functions, you can gain a deeper understanding of the world around you and make more informed decisions.