# 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.

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### 2 Answers

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.**

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.