Complete the following steps for each function.

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Question

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={x^3+4x+1 if x≤0

2x^3 if x>0; a=0

Accepted Answer

Welcome back, everyone. Determine the intervals of continuity for the piece wise function G of X defined as G of X equals 2 X2 minus 5 if X is less than 2, and 3 minus 4 X if X is greater than or equal to 2. And were given for answer choices from negative infinity to 2 B from 2 to infinity C from negative infinity to minus 2 and from 2 to infinity and Dsus from minus infinity to 2 and from 2 to infinity. All of those are open intervals, right? So essentially, let's consider this piece wise function. It consists of two functions. The first one is Y equals 2x2 minus 5. It is a polynomial function that is defined for all real numbers, right? So essentially it is defined for all x values, meaning we are not going to have any discontinuity points inside of it, right? So X belongs to all real numbers. The other function is 3 minus 4 X. Once again it is a polynomial function. The main difference is that the first one is a parabola and the second one is a linear function, and this function is also defined for all real numbers, meaning for this piece-wise function we are only considering one point of discontinuity, which is where the function change changes behavior. And that point is x equals 2 because this is where the function changes behavior from a parabola to a linear function. So our point of interest is a equals 2. In order to have a continuous function, the limit. As X approaches 2 from the left of G of X. Must be equal to G at. 2. That's because 2 X2 minus 5 is applicable if X is less than 2, right? So that means we must approach 2 from the left, and the value that we get for the limit must be equal to the value of the function at 2. And now let's apply this definition. If this definition holds, then the function is continuous at 0.2. Let's evaluate the limit. Limit as X approaches 2 from the left of G of X. Would correspond to the limit of 2 x 2 minus 5. And we can apply the right substitution that's 2 multiplied by 22 minus 5. What do we get? 2 multiplied by 4 gives us 8 and 8 minus 5 gives us 3. So the limit is 3, but what about the value of the function? What about G at 0.2? For this purpose, we need to use the linear function because the G of X function is defined as 2 via the linear form. So that's 3 minus 4 multiplied by 2, and this gives us 3 minus 8, which is -5. Now are they equal? We is not equal to -5, so the limit is not equal to the value of the function at that point, meaning. A equals 2 is a point of discontinuity. So we have to take a set of all real numbers and eliminate. 2, which means that first of all we are taking an interval from negative infinity up to our discontinuity, which is 2, and then from the point of discontinuity up to infinity. And now if we look at the answer choices, well, we can conclude that the correct answer choice to this problem is option D. Thank you for watching.

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={x^3+4x+1 if x≤0
2x^3 if x>0; a=0

Key Concepts

Continuity of Functions

Piecewise Functions

Limits

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