Which Of The Following Accurately Lists All Discontinuities Of The Given Function?$[ F(x) = \begin{cases} 4, & X \ \textless \ -4 \ (x+2)^2, & -4 \leq X \leq -2 \ -\frac{1}{2}x + 1, & -2 \ \textless \ X \ \textless \ 4 \ -1, & X \

Understanding Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These sub-functions are often referred to as "pieces" of the function, and they are combined to form a single function. Piecewise functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions.

The Given Function

The given function is defined as:

${ f(x) = \begin{cases} 4, & x \ \textless \ -4 \\ (x+2)^2, & -4 \leq x \leq -2 \\ -\frac{1}{2}x + 1, & -2 \ \textless \ x \ \textless \ 4 \\ -1, & x \geq 4 \end{cases} }$

Identifying Discontinuities

A discontinuity in a function occurs when the function is not continuous at a particular point. In other words, a discontinuity occurs when the function's graph has a gap or a jump at a certain point. To identify the discontinuities of the given function, we need to examine the function's behavior at each of the interval boundaries.

Interval Boundary 1: x = -4

At x = -4, the function changes from the constant function f(x) = 4 to the quadratic function f(x) = (x+2)^2. This change in function definition can potentially introduce a discontinuity at x = -4.

Interval Boundary 2: x = -2

At x = -2, the function changes from the quadratic function f(x) = (x+2)^2 to the linear function f(x) = -\frac{1}{2}x + 1. This change in function definition can potentially introduce a discontinuity at x = -2.

Interval Boundary 3: x = 4

At x = 4, the function changes from the linear function f(x) = -\frac{1}{2}x + 1 to the constant function f(x) = -1. This change in function definition can potentially introduce a discontinuity at x = 4.

Analyzing the Function's Behavior

To determine whether a discontinuity exists at each of the interval boundaries, we need to examine the function's behavior at those points.

• At x = -4, the function f(x) = (x+2)^2 is equal to 16, which is not equal to the value of the function f(x) = 4 at x = -4. This indicates that there is a discontinuity at x = -4.
• At x = -2, the function f(x) = (x+2)^2 is equal to 0, which is equal to the value of the function f(x) = -\frac{1}{2}x + 1 at x = -2. This indicates that there is no discontinuity at x = -2.
• At x = 4, the function f(x) = -\frac{1}{2}x + 1 is equal to -5, which is not equal to the value of the function f(x) = -1 at x = 4. This indicates that there is a discontinuity at x = 4.

Conclusion

Based on the analysis, the function f(x) has discontinuities at x = -4 and x = 4.

Accurate Listing of Discontinuities

The accurate listing of discontinuities of the given function is:

• x = -4
• x = 4

Note

The function f(x) has no discontinuities at x = -2.

Additional Considerations

When working with piecewise functions, it's essential to consider the function's behavior at each of the interval boundaries. This includes examining the function's values at those points and determining whether the function is continuous or discontinuous at each of those points.

Conclusion

Frequently Asked Questions

Q: What is a discontinuity in a function?

A: A discontinuity in a function occurs when the function is not continuous at a particular point. In other words, a discontinuity occurs when the function's graph has a gap or a jump at a certain point.

Q: How do I identify discontinuities in a piecewise function?

A: To identify discontinuities in a piecewise function, you need to examine the function's behavior at each of the interval boundaries. This includes examining the function's values at those points and determining whether the function is continuous or discontinuous at each of those points.

Q: What are some common types of discontinuities?

A: Some common types of discontinuities include:

• Jump discontinuities: These occur when the function's graph has a jump or a gap at a certain point.
• Infinite discontinuities: These occur when the function's graph has a vertical asymptote at a certain point.
• Removable discontinuities: These occur when the function's graph has a hole or a gap at a certain point.

Q: How do I determine whether a discontinuity is removable or non-removable?

A: To determine whether a discontinuity is removable or non-removable, you need to examine the function's behavior at the point of discontinuity. If the function's limit exists at the point of discontinuity, then the discontinuity is removable. If the function's limit does not exist at the point of discontinuity, then the discontinuity is non-removable.

Q: Can a function have multiple discontinuities?

A: Yes, a function can have multiple discontinuities. In fact, a piecewise function can have multiple discontinuities at different interval boundaries.

Q: How do I graph a piecewise function with discontinuities?

A: To graph a piecewise function with discontinuities, you need to graph each of the individual functions that make up the piecewise function. You should also indicate the points of discontinuity on the graph.

Q: Can a function be continuous at a point and still have a discontinuity at that point?

A: No, a function cannot be continuous at a point and still have a discontinuity at that point. If a function is continuous at a point, then it must be continuous at that point.

Q: How do I determine whether a function is continuous or discontinuous at a point?

A: To determine whether a function is continuous or discontinuous at a point, you need to examine the function's behavior at that point. If the function's limit exists at the point, then the function is continuous at that point. If the function's limit does not exist at the point, then the function is discontinuous at that point.

Q: Can a function have a discontinuity at a point and still be continuous at that point?

A: No, a function cannot have a discontinuity at a point and still be continuous at that point. If a function has a discontinuity at a point, then it must be discontinuous at that point.

Q: How do I find the limit of a function at a point of discontinuity?

A: To find the limit of a function at a point of discontinuity, you need to examine the function's behavior at that point. If the function's limit exists at the point, then you can find the limit by evaluating the function at that point. If the function's limit does not exist at the point, then the limit does not exist.

Q: Can a function have a discontinuity at a point and still have a limit at that point?

A: No, a function cannot have a discontinuity at a point and still have a limit at that point. If a function has a discontinuity at a point, then the limit does not exist at that point.

Q: How do I determine whether a function is differentiable at a point of discontinuity?

A: To determine whether a function is differentiable at a point of discontinuity, you need to examine the function's behavior at that point. If the function's derivative exists at the point, then the function is differentiable at that point. If the function's derivative does not exist at the point, then the function is not differentiable at that point.

Q: Can a function be differentiable at a point and still have a discontinuity at that point?

A: No, a function cannot be differentiable at a point and still have a discontinuity at that point. If a function is differentiable at a point, then it must be continuous at that point.

Q: How do I find the derivative of a function at a point of discontinuity?

A: To find the derivative of a function at a point of discontinuity, you need to examine the function's behavior at that point. If the function's derivative exists at the point, then you can find the derivative by evaluating the function at that point. If the function's derivative does not exist at the point, then the derivative does not exist.

Q: Can a function have a discontinuity at a point and still have a derivative at that point?

A: No, a function cannot have a discontinuity at a point and still have a derivative at that point. If a function has a discontinuity at a point, then the derivative does not exist at that point.

Q: How do I determine whether a function is integrable at a point of discontinuity?

A: To determine whether a function is integrable at a point of discontinuity, you need to examine the function's behavior at that point. If the function's integral exists at the point, then the function is integrable at that point. If the function's integral does not exist at the point, then the function is not integrable at that point.

Q: Can a function be integrable at a point and still have a discontinuity at that point?

A: No, a function cannot be integrable at a point and still have a discontinuity at that point. If a function has a discontinuity at a point, then the integral does not exist at that point.

Q: How do I find the integral of a function at a point of discontinuity?

A: To find the integral of a function at a point of discontinuity, you need to examine the function's behavior at that point. If the function's integral exists at the point, then you can find the integral by evaluating the function at that point. If the function's integral does not exist at the point, then the integral does not exist.

Q: Can a function have a discontinuity at a point and still have an integral at that point?

A: No, a function cannot have a discontinuity at a point and still have an integral at that point. If a function has a discontinuity at a point, then the integral does not exist at that point.

Q: How do I determine whether a function is a solution to a differential equation at a point of discontinuity?

A: To determine whether a function is a solution to a differential equation at a point of discontinuity, you need to examine the function's behavior at that point. If the function satisfies the differential equation at the point, then the function is a solution to the differential equation at that point. If the function does not satisfy the differential equation at the point, then the function is not a solution to the differential equation at that point.

Q: Can a function be a solution to a differential equation at a point and still have a discontinuity at that point?

A: No, a function cannot be a solution to a differential equation at a point and still have a discontinuity at that point. If a function has a discontinuity at a point, then the function is not a solution to the differential equation at that point.

Q: How do I find the solution to a differential equation at a point of discontinuity?

A: To find the solution to a differential equation at a point of discontinuity, you need to examine the function's behavior at that point. If the function satisfies the differential equation at the point, then you can find the solution by evaluating the function at that point. If the function does not satisfy the differential equation at the point, then the solution does not exist at that point.

Q: Can a function have a discontinuity at a point and still have a solution to a differential equation at that point?

A: No, a function cannot have a discontinuity at a point and still have a solution to a differential equation at that point. If a function has a discontinuity at a point, then the solution does not exist at that point.

Q: How do I determine whether a function is a solution to an integral equation at a point of discontinuity?

A: To determine whether a function is a solution to an integral equation at a point of discontinuity, you need to examine the function's behavior at that point. If the function satisfies the integral equation at the point, then the function is a solution to the integral equation at that point. If the function does not satisfy the integral equation at the point, then the function is not a solution to the integral equation at that point.

Q: Can a function be a solution to an integral equation at a point and still have a discontinuity at that point?

A: No, a function cannot be a solution to an integral equation at a point and still have a discontinuity at that point. If a function has a discontinuity at a point, then the function is not a solution to the integral equation at that point.

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