Question

determine the values of b and x so that the following function is continuous on the entire line f(x){x+1 1 13 years ago

Answer 1

sorry typo determine the values of b and c

Answer 2

is it
\[f(x) = \left\{\begin{array}{rcc}
x+1 & \text{if} & 1<x<3 \\
x^2+bx+c& \text{if} & |x-2|\geq 1
\end{array}
\right.
\]

Answer 3

yes

Answer 4

if so it is easier than you think
make sure the two functions agree at \(x=1\) and at \(x=3\)

Answer 5

what do you mean? make them equal or that they are continuous

Answer 6

at \(x=1\) you get \(1+1=2\) for the first expression and \(1+a+b\) for the second, so you know
\[1+a+b=2\] i.e. \[a+b=1\]

Answer 7

at \(x=3\) you get \(3+1=4\) for the top one and \[9+3a+b\] for the bottom one, so you know
\[4=9+3a+b\] or
\[3a+b=-5\]

Answer 8

how do you know 1+a+b=2

Answer 9

sorry i used \(a\) and \(b\) instead of \(b\) and \(c\) but no matter, solve
\[b+c=1\]
\[3b+c=-5\] and you will have your answer

Answer 10

because if you use the top formula \(x+1\) if you replace \(x\) by \(1\) you get \(2\)

Answer 11

all you are really trying to do is make the two expressions agree at 1 and 3 where it changes definition
that is the whole idea behind the problem

Answer 12

oh i see

Answer 13

the two expression are \(x+1\) and \(x^2+bx+c\) if you replace \(x\) by \(1\) in both and set them equal you get \(1+1=1+b+c\)

Answer 14

thank you so much

Answer 15

yw