Question

find the domain of f(x)=log(x^2-5x+6)

Answer 1

your answer is in attachment ...
I think so ....

Answer 2

The domain of

$f(x) = log( {x}^{2} - 5x + 6 )$

is the set D= {x: x ∈ (-∞, 2) ∪ (3, ∞)}

• Now, We have,

$f(x) = log( {x}^{2} - 5x + 6 )$

• The logarithmic function is defined only for positive real numbers therefore,

${x}^{2} - 5x + 6 > 0$

${x}^{2} - 3x - 2x + 6 > 0$

$x(x - 3) - 2(x - 3) > 0$

$(x - 2)(x - 3) > 0$

Now for (x - 2)(x - 3) to be positive there will be two cases:

• Case 1 :-

$(x - 2) > 0 \: and \: (x - 3) > 0$

which gives

$x > 3 \: and \: x > 2$

that is

$x > 3 \: or \: x ∈ (3, ∞) \: \: \: - (1)$

• Case 2 :-

$(x - 2) < 0 \: and \: (x - 3) < 0$

which gives

$x < 2 \: and \: x < 3$

that is

$x < 2 \: or \: x ∈( - ∞,2) \: \: \: \: - (2)$

• Now combining (1) and (2), we get the domain of f(x) as set D such that

D= {x: x ∈ (-∞, 2) ∪ (3, ∞)}