What is the domain of   (x-7)/(x^2+3x+8)? I can't figure it out, thanks.

Expert Answers
baxthum8 eNotes educator| Certified Educator

The domain of a function is the set of all possible inputs, or values for which x can be.

Since your function:  `(x - 7) / (x^2+3x+8)`

has a denominator, the denominator can never equal zero.  Therefore we want to find the values at which `x^2 + 3x + 8 = 0.`

Since we can't factor, we could use quadratic formula or complete the square to solve for x.

Quadratic formula:  `(-3+-sqrt(3^2-4*1**8))/2`

Next, in simplifing we get:  `(-3+-sqrt(41))/2`

So, domain is all real #'s except `(-3+-sqrt(41))/2`

or domain is:  `x!= (-3+-sqrt(41))/2`

or `x!= 1.702 or x!= -4.702`

baxthum8 eNotes educator| Certified Educator

Should not be 41, I placed (-8) for some reason.  Should be -23.  Which is not a real solution.  Domain would be all real #'s

aruv | Student

Let

`f(x)=(x-7)/(x^2+3x+8)` .Thus domain of function f(x)  is the value of all x where f(x) is defined.

F(x) is not defined if `x^2+3x+8=0`

i.e.     `x=(-3+-sqrt(3^2-4xx1xx8))/2`

`=(-3+_sqrt(-23))/2=(-3+-sqrt(23)i)/2`   which is not real numbers.

the graph of g(x)=`x^2+3x+8` does not intersect x-axis .It shows that f(x) is defined for real numbers.

Thus domain of function f(x) is all real numbers if f(x) is real function.