The Function { G $}$ Is Given By { G(x) = 3 \csc (\pi(x+2)) - 1 $}$.Which Of The Following Describes The Range Of { G $}$?A. The Range Of { G $}$ Is { [-3, 3]$}$.B. The Range Of { G $}$

The function g(x) is given by the equation g(x) = 3csc(π(x+2)) - 1. This equation involves the cosecant function, which is the reciprocal of the sine function. To determine the range of g(x), we need to understand the behavior of the cosecant function and how it affects the overall function g(x).

Understanding the Cosecant Function

The cosecant function, denoted by csc(x), is defined as the reciprocal of the sine function: csc(x) = 1/sin(x). The range of the cosecant function is all real numbers except for zero, since the sine function can take on any value between -1 and 1, and the reciprocal of these values will be all real numbers except for zero.

Analyzing the Function g(x)

The function g(x) is given by g(x) = 3csc(π(x+2)) - 1. To determine the range of g(x), we need to consider the behavior of the cosecant function within the argument π(x+2). Since the argument of the cosecant function is π(x+2), we need to consider the values of x that will result in the argument being equal to zero.

Finding the Values of x That Result in Zero

The argument of the cosecant function, π(x+2), will be equal to zero when x+2 = 0. Solving for x, we get x = -2. This means that when x = -2, the argument of the cosecant function will be equal to zero, and the function g(x) will be undefined at this point.

Determining the Range of g(x)

Since the argument of the cosecant function is π(x+2), we need to consider the values of x that will result in the argument being equal to zero. As we have already determined, this occurs when x = -2. However, since the function g(x) is undefined at this point, we need to consider the behavior of the function as x approaches -2.

As x approaches -2, the argument of the cosecant function approaches zero, and the function g(x) approaches negative infinity. Similarly, as x approaches -1, the argument of the cosecant function approaches π, and the function g(x) approaches 3.

Conclusion

Based on the analysis of the function g(x), we can conclude that the range of g(x) is all real numbers except for -3 and 3. This is because the function g(x) approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

The Correct Answer

The correct answer is A. The range of g(x) is [-3, 3].

Range of g(x)

The range of g(x) is all real numbers except for -3 and 3.

Graph of g(x)

The graph of g(x) is a continuous function that approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

Key Takeaways

• The function g(x) is given by g(x) = 3csc(π(x+2)) - 1.
• The range of g(x) is all real numbers except for -3 and 3.
• The function g(x) approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

Final Thoughts

Q: What is the function g(x) and how is it defined?

A: The function g(x) is defined as g(x) = 3csc(π(x+2)) - 1. This equation involves the cosecant function, which is the reciprocal of the sine function.

Q: What is the range of the cosecant function?

A: The range of the cosecant function is all real numbers except for zero. This is because the sine function can take on any value between -1 and 1, and the reciprocal of these values will be all real numbers except for zero.

Q: How does the argument of the cosecant function affect the range of g(x)?

A: The argument of the cosecant function, π(x+2), will be equal to zero when x+2 = 0. Solving for x, we get x = -2. This means that when x = -2, the argument of the cosecant function will be equal to zero, and the function g(x) will be undefined at this point.

Q: What happens to the function g(x) as x approaches -2?

A: As x approaches -2, the argument of the cosecant function approaches zero, and the function g(x) approaches negative infinity.

Q: What happens to the function g(x) as x approaches -1?

A: As x approaches -1, the argument of the cosecant function approaches π, and the function g(x) approaches 3.

Q: What is the range of g(x)?

A: The range of g(x) is all real numbers except for -3 and 3.

Q: Why is the range of g(x) restricted to all real numbers except for -3 and 3?

A: The range of g(x) is restricted to all real numbers except for -3 and 3 because the function g(x) approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

Q: Can you provide a graph of g(x)?

A: The graph of g(x) is a continuous function that approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

Q: What are some key takeaways from this article?

A: Some key takeaways from this article are:

• The function g(x) is given by g(x) = 3csc(π(x+2)) - 1.
• The range of g(x) is all real numbers except for -3 and 3.
• The function g(x) approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1.

Q: What is the final thought on this topic?

A: In conclusion, the function g(x) is a continuous function that approaches negative infinity as x approaches -2, and approaches 3 as x approaches -1. The range of g(x) is all real numbers except for -3 and 3. This is a classic example of a function that has a restricted range due to the behavior of the cosecant function within the argument.