Piecewise defined functionsg(x) is a piecewise defined function such that Ax+B, when x <-1; 2x when    -1 <= x <=2 ; 2Bx-A when x>2; Find the values of A and B to be continous at -1...

Piecewise defined functions

g(x) is a piecewise defined function such that Ax+B, when x <-1; 2x when    -1 <= x <=2 ; 2Bx-A when x>2;

Find the values of A and B to be continous at -1 but discontinous at x=2.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that a function keeps its continuity at a point if side limits are equal, hence the function is continuous at x=-1 if `lim_(x-gt-1,xlt-1) g(x) = lim_(x-gt-1,xgt-1) g(x)`

You need to substitute `Ax+B ` for g(x) in  `lim_(x-gt-1,xlt-1) g(x)`  and `2x ` for g(x) in  `lim_(x-gt-1,xgt-1) g(x)`  such that:

`lim_(x-gt-1,xlt-1) Ax+B = lim_(x-gt-1,xgt-1) 2x`

You need to substitute -1 for x such that:

`-A + B = -2`

You need to remember that a function does not keep its continuity at a point if side limits are not equal such that:

`lim_(x-gt-1,xlt2) g(x) != lim_(x-gt-1,xgt2) g(x)`

`lim_(x-gt-1,xlt2) 2x != lim_(x-gt-1,xgt2) 2Bx - A`

You need to substitute 2 for x such that:

`4 != 4B - A =gt A != 4B - 4`

`A = B+2`

You need to substitute B+2 for A in `A != 4B - 4`  such that:

`B+2 != 4B - 4 =gt 4B-B != 4+2 =gt 3B != 6 =gt B != 2`

`A != 4*2 - 4 =gt A !=4 `

Hence, evaluating the values of `A`  and `B`  yields that they may be any real value except `4`  for `A`  and `2`  for `B` .

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