Find the Constant a Such That the Function is Continuous on the Entire Number Line 3x3 X 1
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{5 Pts.} Find the constant a, or the constant a and b, such that the function is 3x3_ x <1) continuous on the entire real number line: f (x) = {ax + 5,x >1)
Related Question
Find the constant $a$, or the constants $a$ and $b$, such that the function is continuous on the entire real number line. $$f(x)=\left\{\begin{array}{ll} 3 x^{2}, & x \geq 1 \\ a x-4, & x<1 \end{array}\right.$$
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Video Transcript
were given a piecewise function and we're asked to find of the value of A. Is not defined. We know it's a constant, but the value of A. That would make the entire piecewise function continuous for all real numbers. So the functions consisted of two sub functions, the first one being three X squared and we know three X squared is just a problem that's been vertically stretched. So that's going to be defined for all ruined values. It's going to be one continuous function. So we're good when X is greater than or equal to one or not really equal to one, but when X is greater than one and you'll see why, then we also have this a X minus four. And even though we don't know what A is, we know this could be a general form for a first degree uh polynomial and first degree polynomial are going to be continuous for all real numbers on their domain. So we're also good when X is less than one. And you might ask, why didn't we include the one when it's X. Is greater than one on the problem? Well, that's because it could be defined at X is greater than one. But we could have a a graph or yeah, we could see the graph could be continuous until we get to a certain point. And then even that point could be defined. But it might pick up over here and when it picks up over here, it could be at a lower point or a higher point. And then it could continue and be a continuous graph. But we would still have a point this continuity and that's why the X equals to one. That's why we have to check their that it continues because we want to make sure that we have a graph that goes and then is then defined and then continues from that point. That we don't have any breaks in the graph. That's we're trying to get here. So we need to make sure that X. Is equal to one that these two values equal seem. So when we plug in three X squared, we plug in one for X. We get three one squared. And that's going to give us The value of three. Now this must be equal to three. So AX -4 -4 Needs to equal to three. So we add four to both sides. Let me let me go down a little bit. We had a four to both sides. We get A X. Is equal to seven. X. Is just going to equal to one for this part. So eight times one is equal to seven divided by one on each side. And we see that A is equal to seven. So that's how you solve for this problem. And now we have a continuous function that is continuous on the entire number real number line. Or for all real numbers.
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