Question

Pre-Cal Help!!
Find domain of f:
f(x)=(sqrt(2x-5))/(x^2-5x+4)
The answer is [5/2, 4) U (4, infinity)
I understand how to get the 5/2but where does the 4 come from? And It's been a yr and a half since I did college algebra, can someone pls re-explain how you get the bracket and parentheses? TYSM for any help!

Answer 1

To find the domain, you need to EXCLUDE points which make DENOMINATOR =0
so, here, first you'll solve for
\(x^2-5x+4=0\)
which gives you
\(x=4,1\)
so, you have to exclude points 4 and 1 from the domain.
note that, since you have already concluded \(x \ge 5/2\) (right?)
excluding x=1 is unnecessary.
so, you just need to exclude x=4 from the domain you got.
from \(x \ge 5/2\) , domain\( =[5/2, \infty) \)
but 4 cannot lie inside the domain of f(x)
so we break this domain into 2 sub-domains and then take its union.
First from 5/2 to 4, then from 4 to infinity
so, final domain \(=[5/2, 4) U (4, \infty) \)

now about brackets, [ and ] brackets are used when x can take that particular value (domain contains that value)
here, 5/4 is included in domain, so there's a [ bracket in front of it.
but 4 isn't included, so there are ( and ) brackets while writing the domain.
does this make sense ??
ask if still any doubts.....

Answer 2

Understood everyone clearly .?

Answer 3

*everything

Answer 4

why do we not exclude x=1?

Answer 5

because denominator can't be 0
it doesn't exist in the defenition

Answer 6

we did exclude x=1
when we say \(x \ge 5/2 ,\: x \ge 2.5 \implies x\ne 1 \)
thats why we didn't have to exclude it again....

Answer 7

That's right! I understand now! I really needed to refresh my mind on the algebra in here. Thank you so much. One more question...When will x ever be less than a value?

Answer 8

For expressions like \(\sqrt{x-a}\), since term inside square root must be >= 0
\(x-a \ge 0 \implies x\ge a \)
here we get x greater than a value.
to get less than value, you just need an expression like \(\sqrt{a-x}\)
and again since term inside square root must be >= 0
\(a-x \ge 0 \implies x\le a \)

Answer 9

I see! Thank you for the explanations! You really helped.