Question

Not sure how to find a and b?

Answer 1

First let see if we can simplify the piecewise.

`\frac{x {}^(2) - 4 }{x - 2} = ((x + 2)(x - 2))/(x - 2) = (x + 2)`

So rewrite the top equation as x+2.

Second, let evaluate the equation.

For a function to be continuous, one sided limits must be equal to each other so

Limit as of x+2 approaches two, must equal the limit as x approaches ax^2-bx-14

We direct subsitue for both functions.

`(2 + 2) = 4`

`a( {2)}^(2) - b(2) - 14`

`4a - 2b - 14 = 4`

`4a - 2b - 18`

Also the limit as x approaches 3 of 10x-a+b must equal the limit of ax^2-bx-14 as x approaches 3.

So we direct subsitue again

`a(3) {}^(2) - b(3) - 14`

`9a - 3b - 14`

`10(3) - a + b`

`30 - a + b`

Set them equal to each other

`9a - 3b - 14 = 30 - a + b`

`10a - 4b - 44`

Now,we have a system of equations

`10a - 4b = 44`

and

`4a - 2b = 18`

Eliminate b variable.

`- 2(4a - 2b = 18) = - 8a + 4b = - 36`

Add both equations so we get

`10a - 4b = 44`

`(10a - 4b = 44 )+ ( - 8a + 4b = - 36)`

`2a = 8`

`a = 4`

Plug back in to solve for b.

`10(4) - 4b = 44`

`40 - 4b = 44`

`- 4b = 4`

`b = - 1`

So a=4 ,b=-1