What is the domain of definition of the function f(x) = log 2 (x^2 - 5x + 6) ?

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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To find the domain of f(x) = log2(x^2-5x+6).

We know the domain of x is the set of values of x for which log2 (x^2-5x+6) is real.

This is when x^2-5x+6 > 0.

Therefore  x^2-5x+ (5/2)^2 -(5/2)^2+6 > 0

Therefore (x-5/2)^2 - 6.25+6 > 0.

(x-5/2)^2 > 0.25.

(x-5/2) > (1/2) , Or x-5/2 < -1/2.

 Or x  > 5/2+1/2 = 3, Or x < 5/2-1/2 = 2.

So If x > 3 , Or x < 2, then x^2-5x+6 > 0, then f(x) is real.

Therefore the domain of  the function f(x)  is  (-infinity , 2)  or (3,  infinity).

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The domain of definition of the given function contains the admissible values of x for the logarithmic function to exist.

We'll impose the constraint for the logarithmic function to exist: the argument of logarithmic function has to be positive.

x^2 - 5x + 6 > 0

We'll compute the roots of the expression:

x^2 - 5x + 6 = 0

We'll apply the quadratic formula:

x1 = [5 +/- sqrt(25 - 24)]/2

x1 = (5+1)/2

x1 = 3

x2 = 2

The expression is positive over the intervals:

(-infinite , 2) U (3 , +infinite)

So, the logarithmic function is defined for values of x that belong to the intervals (-infinite , 2) U (3 , +infinite).

The reunion of intervals represents the domain of definition of the given function  f(x) = log 2 (x^2 - 5x + 6).

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