Question

In this exercise, find the constant a such that the function is continuous on the entire real line.

Answer 1

.....

Answer 2

I don't get it. :P Is there more?

Answer 3

\[f(x)= 3x^2 when x \ge1 ,
ax-4 when x <1\]

Answer 4

ok. just a minute.

Answer 5

Sorry, I couldn't get it to look very good, and I figured it might be easier to understand with at least the proper signs. I solved it and I got a=-1 but I read somewhere that they got a=7 which is why I was confused.

Answer 6

The only point that could be a problem is at x=3 where the step is. At x=3, f(x)=x^3=27. The limit of f as x approaches 3 from the right needs to be the same. 3^2=9. You need a*9=27, a=3

Answer 7

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Answer 8

Okay, so I kind of get that, but why wouldn't you use 1, since that is mentioned in the piecewise?

Answer 9

Because the way I solved this was looking at whether or not the lim 3x^2 as x approaches 1 from the left is the same as lim ax-4 as x approaches 1 from the right

Answer 10

hmmm...

Answer 11

Since in order for a function to be continuous both functions must be equivalent (meaning that they meet at the same point)

Answer 12

you're right

Answer 13

And from there, substitution could be done to find the a value so that it equals the same as what the limit of 3x^2 is

Answer 14

I think you lost me on that one :P

Answer 15

wait I get it!

Answer 16

I think you were right the 1st time...

Answer 17

So then the answer would be a=-1 because this causes the two limits to be equivalent?

Answer 18

There's a little problem in that the back of the book says that the answer is indeed 7, which doesn't quite make sense for me. However, another problem I did, I got the answer right by the same process.

Answer 19

I see what I did wrong. I thought it was +4 instead of -4