Exercises 5–10 refer to the function
f(x) = { x² − 1, −1...

Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

At what values of x is f continuous?

Accepted Answer

Welcome back, everyone. Consider the graph of the given function G of X. Determine the points at which the function G of X is continuous. Here we have our graph, and for our answer choices, A says it's continuous for all values of X on the interval open parentheses -2 6 closed parentheses except at X equals 0 and 2. B says it's continuous for all values of X and the integral except at X equals 2. C says it is except at X equals 2 and 4, and D also says it is except at 02, and 4. Now, all we're gonna figure out the points at which our function is continuous? Well, if we look at our graph here, the function appears continuous for most parts, meaning there are no breaks or jumps in those regions except at the following values of X. These values look a bit, it looks like we may have to, or they look a bit questionable here. That is where X equals -2, where X equals 0, where X equals 2. And where X equals 4. So let's look at each of these to see if they are continuous at those points, OK? Now, not here. That if there is a jump or a break, it's not continuous where x equals -2. Notice that it's a solid dot, so that means there is no jump or break. Thus it is continuous. Where X equals 0, OK. Notice that there is an open dot, OK, or an empty circle, which means that there is a break. So that tells us then that at X equals 0, OK, the function is discontinuous. Function is discontinuous. Next, if we go to where X equals 2, notice there that we have an open dot where our function is heading to, OK? But we have a solid dot at we're at the 0.24. OK. So what this means is that our function is also discontinuous here because there is a break. So it's discontinuous at X equals 2, OK. And if we go to where X equals 4 again, it's an open dot. So we know it's also discontinuous at X equals 4. Thus, our function is continuous at all values of X and the interval from -2 to 6, except that X equals 02, and 4. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
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At what values of x is f continuous?

Key Concepts

Continuity of Functions

Piecewise Functions

Limits

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