Where is this piecewise function discontinuous?

In summary, g(x) has three different definitions depending on the value of x. For x < -4, g(x) = 3. For |x| ≤ 4, g(x) = 7+x. And for x > 4, g(x) = x^4. There is a jump discontinuity at x=4, meaning that g(x) is not continuous at this point. The limit of g(x) is 256 as x approaches 4 from the left.
  • #1
Ianfinity
5
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g(x) = 3 if x < -4
g(x) = 7+x if |x| <or= 4
g(x) = x^4 if x > 4


I know there is a jump discontinuity at x=4. How would I state that in interval notation? Is that even possible or is it good enough to say g(x) is discontinuous at x=4? Basically what I've found is that if x=4 then g(x)=11 and that the limit of x is 256 as x approaches 4 from the right.

This is part of my review for the test I will be having today. Thanks in advance for the help.
 
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  • #2
Ianfinity said:
g(x) = 3 if x < -4
g(x) = 7+x if |x| <or= 4
g(x) = x^4 if x > 4

I know there is a jump discontinuity at x=4. How would I state that in interval notation? Is that even possible or is it good enough to say g(x) is discontinuous at x=4? Basically what I've found is that if x=4 then g(x)=11 and that the limit of x is 256 as x approaches 4 from the right.

This is part of my review for the test I will be having today. Thanks in advance for the help.
Hello Ianfinity. Welcome to PF !

Yes, g(x) is discontinuous at x=4.

The limit of g(x) is 256 as x → 4-,

not: the limit of x is 256 as x → 4-.
 

FAQ: Where is this piecewise function discontinuous?

What is a piecewise function?

A piecewise function is a mathematical function that is defined by different equations on different parts of its domain. This means that the function may have different rules for different input values.

How can I identify where a piecewise function is discontinuous?

A piecewise function is discontinuous at any point where there is a "break" in the function, meaning that the function has a different value on either side of that point. This can be identified by looking at the graph of the function or by evaluating the function at different values in the domain.

What causes a piecewise function to be discontinuous?

A piecewise function can be discontinuous for a variety of reasons. These can include a "jump" in the function where the value changes suddenly, a vertical asymptote where the function approaches infinity, or a removable discontinuity where the function has a hole or gap at a certain point.

How do discontinuities in a piecewise function affect its graph?

Discontinuities in a piecewise function can cause breaks or gaps in the graph, as the function may have different values on either side of the discontinuity. This can result in a non-smooth or jagged graph with sharp turns or sudden changes in direction.

Are there any strategies for graphing piecewise functions and identifying discontinuities?

Yes, there are a few strategies that can be helpful when graphing piecewise functions and identifying discontinuities. These include looking for patterns in the equations that make up the function, evaluating the function at key points, and using a graphing calculator to visualize the function and its discontinuities.

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