If f(x) is discontinuous, determine the reason. f of x equals the quantity x squared plus 4 for x less than or equal to 1 and equals x plus 4 for x greater than 1 (5 points)
A. f(x) is continuous for all real numbers
B. The limit as x approaches 1 does not exist
C. f(1) does not equal the limit as x approaches 1
D. f(1) is not defined

 Choice A

f(x) is continuous for all real numbers

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Explanation:

The piecewise function given to us basically breaks down into these two cases:

So if we plug in x = 1, then we go for the top line and say

f(x) = x^2+4

f(1) = 1^2+4

f(1) = 5

Showing that f(1) is indeed defined. We rule out choice D because of this.

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Let's see what happens when we plug x = 1 into the bottom equation

f(x) = x+4

f(1) = 1+4

f(1) = 5

Both pieces produce the same output. This tells us that as x approaches 1 from either the left or right sides, the y value approaches y = 5. Therefore, the limit at x = 1 exists. We rule out choice B because of this.

We rule out choice C as well because f(1) does equal the limiting value mentioned earlier.

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The answer is choice A because no matter what x value we plug in, it's not only defined but also it's connected to the rest of the function curve as shown below.

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my guess would be b and d, but i didn’t actually get out the pen and paper. lol