f(x)= x^2 - 4x + 5 , x > 2 a) Express f(x) in the form (x+b)^2 + c b) State the range of f c) Find an expression for f^-1(x) and state its domain ^2 means squared and x > 2 should be 'x is greater than or equal to 2'

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f(x)= x^2 - 4x + 5 , x > 2

a) Express f(x) in the form (x+b)^2 + c

Let us rewrite :

f(x) = x^2 - 4x + 4 + 1 = (x-2)^2 + 1

b) State the range of f

f(x) = x^2 - 4x + 5

==> f'(x)...

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f(x)= x^2 - 4x + 5 , x > 2

a) Express f(x) in the form (x+b)^2 + c

Let us rewrite :

f(x) = x^2 - 4x + 4 + 1 = (x-2)^2 + 1

b) State the range of f

f(x) = x^2 - 4x + 5

==> f'(x) = 2x - 4 = 0

==> x= 2

Then 2 is critical value for f, where fa has a minimun.

Then f(2)  = 1

when x < 2 ==> f  < 1

when x > = 2 ==> f >= 1

 

c) Find an expression for f^-1(x) and state its domain

Let f(x) = y = x^2 - 4x + 5

==> y = (x-2)^2 + 1

==> y-1 = (x-2)^2

==> sqrt(y-1) = x-2

==> sqrt(y-1) + 2 = x

==> f^-1 = sqrt(x-1) + 2

The domain is x values such that f^-1 is defined.

f^-1 is NOT defined when x-1 < 0 ==> x< 1

Then the domain is x = [ 1, inf)

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