How do you find the values of a and b that make x continuous everywhere?Given:  f(x) =      x2 − 4 x − 2      if x < 2  ax2 − bx + 3      if 2 ≤ x < 3...

How do you find the values of a and b that make x continuous everywhere?

Given: 

f(x) = 

 

  x2 − 4 x − 2      if x < 2  ax2 − bx + 3      if 2 ≤ x < 3  4x − a + b      if x ≥ 3

 

Find the values of a and b that make x continuous everywhere.

Asked on by quietbang

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should verify if the function is continuous at x=3 such that:

`lim_(x->3) f(x) = f(3) `

The function has limit at x = 3 if `lim_(x->3,x<3) (x^2-4)/(x-2) = lim_(x->3,x>3)(4x-a+b)` `lim_(x->3,x<3) (x^2-4)/(x-2) = lim_(x->3,x<3) ((x-2)(x+2))/(x-2)`

`lim_(x->3,x<3) (x^2-4)/(x-2) = lim_(x->3,x<3) (x+2) = 3+2 = 5`

`lim_(x->3,x>3)(4x-a+b) = 4*3 - a + b = 12 - a+b`

`-a+b = 5 - 12 => b-a = -7`

Hence, using the condition for a function to be continuous yields a relation between a and b such that b-a=-7.

quietbang's profile pic

quietbang | Student, College Freshman | (Level 2) eNoter

Posted on

That's x^2-4/x-2 if x<2, 

ax^2-bx+3 if 2=<x<3

4x-a+b if x>=3

 

Sorry about the mistakes in writing. 

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