Question

Find `a` and `b` so that `f(x) = { (x^2, xle1),(ax+b,x gt 1) :}` is continuous?

Answer 1

`a=1-b`

Explanation:

We have:

`f(x) = { (x^2, xle1),(ax+b,x gt 1) :}`

Both `x^2` and `ax+b` are polynomials, and so are continuous in there own right, so the only possibility of a discontinuity is at the junction between the two polynomials at `x=1`

In order for continuity at `x=1` then we require (by the definition of continuity):

`lim_(x rarr 1) f(x) = f(1)`

Consider the LH limit;

`lim_(x rarr 1^-) f(x) = lim_(x rarr 1^-) x^2 = 1`

And the RH limit:

`lim_(x rarr 1^+) f(x) = lim_(x rarr 1^+) ax+b =a+b`

So the requirement for continuity is:

`lim_(x rarr 1^-) f(x) = lim_(x rarr 1^+) f(x)`

`:. a+b =1 => a=1-b`