Question

find the constants a and b such that the function is continuous on the entire real number line? What does it mean by "constants" in this case?

Answer 1

ill draw out the function in this case

Answer 2

function is?

Answer 3

is it given peicewise?

Answer 4

if so plug in the values at the endpoints and solve for a and b

Answer 5

yess lol

Answer 6

oh ok.

Answer 7

so say it changes at 2
replace x by 2, get an expression in a and b
then if it changes again at 3
replace x by 3
get another expression in a and b
solve the system

Answer 8

go ahead and post, we can work it out if what i wrote is not clear

Answer 9

im trying to write the function but its coming out weird on here today. its basically x^3 if x is less than or equal to 2, and ax^2 if x > 2

Answer 10

in this case you would just plug in 2?

Answer 11

ok so right away we want
\[ 2^3=a\times 2^2\]

Answer 12

just plug in 2 for x. solve for with your eyeballs, get
\[a=2\]

Answer 13

that way the functions will match up at 2. the limit will be 8 from the left and from the right.

Answer 14

then if it changes again at some other place, repeat the process

Answer 15

so if lets say there are 3 parts to the piecewise, all 3 would have to equal to eachother?

Answer 16

well there can't be three parts at one place. there can only be two right? but it can be defined differently at different places, so just plug in the value of x where it changes places into both formulas, set them equal and solve

Answer 17

i saw one like this were you had to actually solve a system
you don't in this case because you know a = 2 right away

Answer 18

f(x)= 2 x\[\le\]-1

ax+b -1<x<3

-2 x\[\ge\] 3

Answer 19

ok this doesnt like me very well, but i tried to write the function -.-

Answer 20

now you get a system i think. replace x by -1 get
\[-2 = -a+b\]

Answer 21

i replaced x by -1 in top two. then replace x by 3 get
\[3a+b = -\]

Answer 22

dang
\[3a+b=-6\]

Answer 23

so solve the system
\[-a+b=-2\]
\[3a+b=-6\] and you will have your a and your b

Answer 24

i gotta run but i hope it is clear what i did. make the replacement for x in both, got two equations for a and b. then solve

Answer 25

okey dokes thank you. I'll try solving this the best i can