The function f is defined by f(x)=(x-1)(3+x), x>b-1 f(x)=-x+1, x<=b-1 and is continuous at b-1. What is b?

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1 Answer
Sep 26, 2017

b=2, or b=-3

Explanation:

We can write the function as follows:

f(x) = { (-x+1, x<=b-1), ((x-1)(3+x), x>b-1) :}

As f is continuous at b-1 then;

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

This means that:

-(b-1)+1 = ((b-1)-1)(3+(b-1))

:. -b+1+1 = (b-2)(b+2)

:. 2-b = b^2-4

:. b^2+b-6 = 0

:. (b -2)(b+ 3)= 0

Hence we have:

b=2, or b=-3