The Function { G $}$ Is Defined Below.${ G(x)=\begin{cases} x-4, & X \leq 1 \ -3, & 1\ \textless \ X \leq 3 \ x-6, & X\ \textgreater \ 3 \end{cases} }$Which Statement Is True?A. The { Y $} − I N T E R C E P T O F \[ -intercept Of \[ − In T Erce Pt O F \[ G
Introduction
In mathematics, functions are used to describe the relationship between variables. A function can be defined in various ways, including algebraic, trigonometric, and piecewise functions. In this article, we will focus on a piecewise function g(x) and explore its properties.
The Function g(x)
The function g(x) is defined as:
This function is a piecewise function, meaning it is defined in different ways for different intervals of x. The function has three different definitions:
- For x ≤ 1, g(x) = x - 4
- For 1 < x ≤ 3, g(x) = -3
- For x > 3, g(x) = x - 6
Understanding the Function
To understand the function g(x), we need to analyze each of its definitions. Let's start with the first definition, g(x) = x - 4, which is valid for x ≤ 1. This definition represents a linear function with a slope of 1 and a y-intercept of -4.
The second definition, g(x) = -3, is valid for 1 < x ≤ 3. This definition represents a constant function with a value of -3.
The third definition, g(x) = x - 6, is valid for x > 3. This definition represents a linear function with a slope of 1 and a y-intercept of -6.
Finding the y-Intercept
The y-intercept of a function is the point where the function intersects the y-axis. In other words, it is the value of the function when x = 0.
To find the y-intercept of g(x), we need to evaluate the function at x = 0. However, we need to be careful because the function g(x) is defined in different ways for different intervals of x.
For x ≤ 1, g(x) = x - 4. Evaluating this definition at x = 0, we get g(0) = 0 - 4 = -4.
For 1 < x ≤ 3, g(x) = -3. Evaluating this definition at x = 0, we get g(0) = -3.
For x > 3, g(x) = x - 6. Evaluating this definition at x = 0, we get g(0) = 0 - 6 = -6.
Which Statement is True?
Now that we have found the y-intercept of g(x) for each of its definitions, we can determine which statement is true.
A. The y-intercept of g(x) is -4. B. The y-intercept of g(x) is -3. C. The y-intercept of g(x) is -6.
Based on our analysis, we can see that the y-intercept of g(x) is -4 for x ≤ 1, -3 for 1 < x ≤ 3, and -6 for x > 3. Therefore, the correct answer is:
A. The y-intercept of g(x) is -4.
Conclusion
In conclusion, the function g(x) is a piecewise function defined in different ways for different intervals of x. We analyzed each of its definitions and found the y-intercept of g(x) for each of them. Based on our analysis, we determined that the correct answer is A. The y-intercept of g(x) is -4.
Final Thoughts
The function g(x) is a complex function with multiple definitions. Understanding each of its definitions is crucial to analyzing the function as a whole. By breaking down the function into its individual components, we can gain a deeper understanding of its properties and behavior.
References
- [1] "Functions" by Khan Academy
- [2] "Piecewise Functions" by Math Open Reference
- [3] "Y-Intercept" by Wolfram MathWorld
Additional Resources
- [1] "Functions" by MIT OpenCourseWare
- [2] "Piecewise Functions" by Purplemath
- [3] "Y-Intercept" by Mathway
Discussion
What do you think about the function g(x)? Do you have any questions or comments about the analysis? Share your thoughts in the discussion section below.
Discussion Section
- Question 1: What is the y-intercept of g(x) for x ≤ 1?
- Answer 1: The y-intercept of g(x) for x ≤ 1 is -4.
- Question 2: What is the y-intercept of g(x) for 1 < x ≤ 3?
- Answer 2: The y-intercept of g(x) for 1 < x ≤ 3 is -3.
- Question 3: What is the y-intercept of g(x) for x > 3?
- Answer 3: The y-intercept of g(x) for x > 3 is -6.
Conclusion
Introduction
In our previous article, we analyzed the function g(x) and its properties. We discussed the function's definition, its behavior for different intervals of x, and its y-intercept. In this article, we will answer some frequently asked questions about the function g(x).
Q1: What is the function g(x)?
A1: The function g(x) is a piecewise function defined as:
Q2: What is the y-intercept of g(x)?
A2: The y-intercept of g(x) is -4 for x ≤ 1, -3 for 1 < x ≤ 3, and -6 for x > 3.
Q3: How do I evaluate the function g(x) at a given point x?
A3: To evaluate the function g(x) at a given point x, you need to determine which definition of the function is valid for that point. If x ≤ 1, use the definition g(x) = x - 4. If 1 < x ≤ 3, use the definition g(x) = -3. If x > 3, use the definition g(x) = x - 6.
Q4: What is the domain of the function g(x)?
A4: The domain of the function g(x) is all real numbers x.
Q5: What is the range of the function g(x)?
A5: The range of the function g(x) is all real numbers y.
Q6: How do I graph the function g(x)?
A6: To graph the function g(x), you need to graph each of its definitions separately. For x ≤ 1, graph the line y = x - 4. For 1 < x ≤ 3, graph the horizontal line y = -3. For x > 3, graph the line y = x - 6.
Q7: What is the difference between the function g(x) and the function f(x)?
A7: The function g(x) and the function f(x) are two different functions. The function g(x) is defined as:
The function f(x) is defined as:
Q8: How do I find the inverse of the function g(x)?
A8: To find the inverse of the function g(x), you need to swap the x and y variables and solve for y. This will give you the inverse function g^(-1)(x).
Q9: What is the derivative of the function g(x)?
A9: The derivative of the function g(x) is:
Q10: How do I use the function g(x) in real-world applications?
A10: The function g(x) can be used in various real-world applications, such as modeling population growth, predicting stock prices, and analyzing data.
Conclusion
In conclusion, the function g(x) is a complex function with multiple definitions. Understanding each of its definitions is crucial to analyzing the function as a whole. By breaking down the function into its individual components, we can gain a deeper understanding of its properties and behavior. We hope this Q&A article has helped you understand the function g(x) and its properties.