The Point { (-3,-5)$}$ Is On The Graph Of A Function. Which Equation Must Be True Regarding The Function?A. { F(-3) = -5$}$ B. { F(-3,-5) = -8$}$ C. { F(-5) = -3$}$ D. { F(-5,-3) = -2$}$
The Point on the Graph of a Function: Understanding the Equation
When a point is on the graph of a function, it means that the coordinates of the point satisfy the equation of the function. In this article, we will explore the concept of a function and how to determine the equation that must be true regarding the function when a point is on its graph.
What is a Function?
A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between variables, where each input corresponds to exactly one output. In other words, a function takes an input and produces a unique output.
The Graph of a Function
The graph of a function is a visual representation of the function, where each point on the graph corresponds to a pair of input and output values. The graph can be thought of as a mapping from the domain to the range, where each point on the graph represents a unique input-output pair.
The Point on the Graph
Given that the point {(-3,-5)$}$ is on the graph of a function, we need to determine which equation must be true regarding the function. To do this, we need to understand the relationship between the point and the function.
Understanding the Equation
The equation of a function is typically written in the form {f(x) = y$}$, where {x$}$ is the input and {y$}$ is the output. When a point is on the graph of a function, it means that the input and output values of the point satisfy the equation of the function.
Analyzing the Options
Let's analyze the options given:
A. {f(-3) = -5$}$
This option states that the input is and the output is . Since the point {(-3,-5)$}$ is on the graph of the function, this option must be true.
B. {f(-3,-5) = -8$}$
This option is incorrect because the input is , which is not a valid input for a function. A function takes a single input and produces a single output.
C. {f(-5) = -3$}$
This option is incorrect because the input is , which is not the input value of the point {(-3,-5)$}$.
D. {f(-5,-3) = -2$}$
This option is incorrect because the input is , which is not a valid input for a function.
Conclusion
Based on the analysis of the options, the correct answer is:
A. {f(-3) = -5$}$
This equation must be true regarding the function because the point {(-3,-5)$}$ is on the graph of the function, and the input and output values of the point satisfy the equation of the function.
Understanding the Concept
The concept of a function and its graph is a fundamental idea in mathematics. Understanding how to determine the equation that must be true regarding a function when a point is on its graph is an important skill to develop. By analyzing the options and understanding the relationship between the point and the function, we can determine the correct equation.
Real-World Applications
The concept of a function and its graph has many real-world applications. For example, in physics, the position of an object as a function of time is a function that can be represented graphically. In economics, the demand for a product as a function of its price is a function that can be represented graphically. In computer science, the output of a program as a function of its input is a function that can be represented graphically.
Conclusion
In conclusion, the point {(-3,-5)$}$ is on the graph of a function, and the equation that must be true regarding the function is {f(-3) = -5$}$. Understanding the concept of a function and its graph is an important skill to develop, and it has many real-world applications. By analyzing the options and understanding the relationship between the point and the function, we can determine the correct equation.
References
- [1] "Functions" by Khan Academy
- [2] "Graphs of Functions" by Math Open Reference
- [3] "Functions and Graphs" by Wolfram MathWorld
Additional Resources
- [1] "Functions and Graphs" by MIT OpenCourseWare
- [2] "Functions and Relations" by University of California, Berkeley
- [3] "Graphs of Functions" by University of Michigan
The Point on the Graph of a Function: Q&A
In our previous article, we explored the concept of a function and its graph, and how to determine the equation that must be true regarding the function when a point is on its graph. In this article, we will answer some frequently asked questions (FAQs) related to the topic.
Q: What is the difference between a function and a relation?
A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between variables, where each input corresponds to exactly one output. A relation, on the other hand, is a set of ordered pairs, where each pair represents a possible input-output combination.
Q: How do I determine if a relation is a function?
A: To determine if a relation is a function, you need to check if each input corresponds to exactly one output. In other words, you need to check if each input has a unique output. If each input has a unique output, then the relation is a function.
Q: What is the domain and range of a function?
A: The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values.
Q: How do I find the domain and range of a function?
A: To find the domain and range of a function, you need to analyze the function and determine the possible input and output values. For example, if the function is {f(x) = x^2$}$, then the domain is all real numbers, and the range is all non-negative real numbers.
Q: What is the graph of a function?
A: The graph of a function is a visual representation of the function, where each point on the graph corresponds to a pair of input and output values.
Q: How do I graph a function?
A: To graph a function, you need to plot the points on the graph that correspond to the input and output values of the function. You can use a graphing calculator or a computer program to graph the function.
Q: What is the equation of a function?
A: The equation of a function is a mathematical expression that describes the relationship between the input and output values of the function.
Q: How do I write the equation of a function?
A: To write the equation of a function, you need to analyze the function and determine the mathematical expression that describes the relationship between the input and output values. For example, if the function is {f(x) = x^2$}$, then the equation of the function is {y = x^2$}$.
Q: What is the point on the graph of a function?
A: The point on the graph of a function is a specific point that corresponds to a pair of input and output values.
Q: How do I determine the equation that must be true regarding the function when a point is on its graph?
A: To determine the equation that must be true regarding the function when a point is on its graph, you need to analyze the point and determine the input and output values that correspond to the point. Then, you need to use the equation of the function to determine the equation that must be true.
Q: What is the significance of the point on the graph of a function?
A: The point on the graph of a function is significant because it represents a specific input-output combination that satisfies the equation of the function.
Q: How do I use the point on the graph of a function to determine the equation that must be true regarding the function?
A: To use the point on the graph of a function to determine the equation that must be true regarding the function, you need to analyze the point and determine the input and output values that correspond to the point. Then, you need to use the equation of the function to determine the equation that must be true.
Conclusion
In conclusion, the point on the graph of a function is a specific point that corresponds to a pair of input and output values. The equation that must be true regarding the function when a point is on its graph can be determined by analyzing the point and using the equation of the function. By understanding the concept of a function and its graph, you can determine the equation that must be true regarding the function when a point is on its graph.
References
- [1] "Functions" by Khan Academy
- [2] "Graphs of Functions" by Math Open Reference
- [3] "Functions and Graphs" by Wolfram MathWorld
Additional Resources
- [1] "Functions and Graphs" by MIT OpenCourseWare
- [2] "Functions and Relations" by University of California, Berkeley
- [3] "Graphs of Functions" by University of Michigan