The Function $g(x$\] Is Defined As Shown:$\[ g(x) = \begin{cases} x-1, & -2 \leq X \ \textless \ -1 \\ 2x+3, & -1 \leq X \ \textless \ 3 \\ 6-x, & X \geq 3 \end{cases} \\]What Is The Value Of $g(3$\]?A. 3 B. 9 C.
Introduction to the Function g(x)
The function g(x) is a piecewise function, which means it is defined differently for different intervals of x. This type of function is commonly used in mathematics to model real-world situations where the behavior of a system or process changes at certain points. In this article, we will explore the function g(x) and its applications in mathematics.
Definition of the Function g(x)
The function g(x) is defined as follows:
{ g(x) = \begin{cases} x-1, & -2 \leq x \ \textless \ -1 \\ 2x+3, & -1 \leq x \ \textless \ 3 \\ 6-x, & x \geq 3 \end{cases} \}
Understanding the Intervals of the Function g(x)
To evaluate the function g(x) at a given value of x, we need to determine which interval it falls into. The intervals are defined as follows:
- For -2 ≤ x < -1, the function is defined as g(x) = x - 1.
- For -1 ≤ x < 3, the function is defined as g(x) = 2x + 3.
- For x ≥ 3, the function is defined as g(x) = 6 - x.
Evaluating the Function g(x) at x = 3
To evaluate the function g(x) at x = 3, we need to determine which interval it falls into. Since x = 3 falls into the interval x ≥ 3, we will use the definition g(x) = 6 - x.
Calculating the Value of g(3)
Using the definition g(x) = 6 - x, we can calculate the value of g(3) as follows:
g(3) = 6 - 3 g(3) = 3
Conclusion
In conclusion, the value of g(3) is 3. This is because x = 3 falls into the interval x ≥ 3, and the function g(x) is defined as g(x) = 6 - x for this interval.
Applications of the Function g(x)
The function g(x) has several applications in mathematics, including:
- Modeling real-world situations where the behavior of a system or process changes at certain points.
- Solving equations and inequalities involving piecewise functions.
- Graphing piecewise functions and analyzing their properties.
Graphing the Function g(x)
To graph the function g(x), we need to plot the different pieces of the function on a coordinate plane. The graph of g(x) will consist of three line segments, each corresponding to one of the intervals defined above.
Analyzing the Properties of the Function g(x)
The function g(x) has several properties that can be analyzed, including:
- The domain and range of the function.
- The continuity and differentiability of the function.
- The behavior of the function at the endpoints of the intervals.
Domain and Range of the Function g(x)
The domain of the function g(x) is the set of all possible input values, which is the union of the three intervals defined above. The range of the function g(x) is the set of all possible output values, which can be determined by analyzing the behavior of the function at the endpoints of the intervals.
Continuity and Differentiability of the Function g(x)
The function g(x) is continuous at all points except at the endpoints of the intervals, where it has a discontinuity. The function g(x) is differentiable at all points except at the endpoints of the intervals, where it has a discontinuity in its derivative.
Behavior of the Function g(x) at the Endpoints of the Intervals
The function g(x) has a discontinuity at the endpoints of the intervals, where the behavior of the function changes abruptly. This can be seen by analyzing the behavior of the function at the endpoints of the intervals.
Conclusion
In conclusion, the function g(x) is a piecewise function that is defined differently for different intervals of x. The function g(x) has several applications in mathematics, including modeling real-world situations, solving equations and inequalities, and graphing piecewise functions. The function g(x) has several properties that can be analyzed, including its domain and range, continuity and differentiability, and behavior at the endpoints of the intervals.
Final Answer
The final answer is:
Frequently Asked Questions About the Function g(x)
Q: What is the function g(x) defined as?
A: The function g(x) is a piecewise function, which means it is defined differently for different intervals of x. It is defined as follows:
{ g(x) = \begin{cases} x-1, & -2 \leq x \ \textless \ -1 \\ 2x+3, & -1 \leq x \ \textless \ 3 \\ 6-x, & x \geq 3 \end{cases} \}
Q: What are the intervals of the function g(x)?
A: The intervals of the function g(x) are defined as follows:
- For -2 ≤ x < -1, the function is defined as g(x) = x - 1.
- For -1 ≤ x < 3, the function is defined as g(x) = 2x + 3.
- For x ≥ 3, the function is defined as g(x) = 6 - x.
Q: How do I evaluate the function g(x) at a given value of x?
A: To evaluate the function g(x) at a given value of x, you need to determine which interval it falls into. Then, you can use the corresponding definition of the function to calculate the value.
Q: What is the value of g(3)?
A: The value of g(3) is 3, because x = 3 falls into the interval x ≥ 3, and the function g(x) is defined as g(x) = 6 - x for this interval.
Q: What are some applications of the function g(x)?
A: The function g(x) has several applications in mathematics, including:
- Modeling real-world situations where the behavior of a system or process changes at certain points.
- Solving equations and inequalities involving piecewise functions.
- Graphing piecewise functions and analyzing their properties.
Q: How do I graph the function g(x)?
A: To graph the function g(x), you need to plot the different pieces of the function on a coordinate plane. The graph of g(x) will consist of three line segments, each corresponding to one of the intervals defined above.
Q: What are some properties of the function g(x)?
A: The function g(x) has several properties that can be analyzed, including:
- The domain and range of the function.
- The continuity and differentiability of the function.
- The behavior of the function at the endpoints of the intervals.
Q: Is the function g(x) continuous at all points?
A: No, the function g(x) is not continuous at all points. It has a discontinuity at the endpoints of the intervals, where the behavior of the function changes abruptly.
Q: Is the function g(x) differentiable at all points?
A: No, the function g(x) is not differentiable at all points. It has a discontinuity in its derivative at the endpoints of the intervals.
Q: What is the domain of the function g(x)?
A: The domain of the function g(x) is the set of all possible input values, which is the union of the three intervals defined above.
Q: What is the range of the function g(x)?
A: The range of the function g(x) is the set of all possible output values, which can be determined by analyzing the behavior of the function at the endpoints of the intervals.
Q: Can I use the function g(x) to model real-world situations?
A: Yes, the function g(x) can be used to model real-world situations where the behavior of a system or process changes at certain points.
Q: Can I use the function g(x) to solve equations and inequalities?
A: Yes, the function g(x) can be used to solve equations and inequalities involving piecewise functions.
Q: Can I use the function g(x) to graph piecewise functions?
A: Yes, the function g(x) can be used to graph piecewise functions and analyze their properties.
Final Answer
The final answer is: