# Thread: determining the largest set D of real numbers

1. ## determining the largest set D of real numbers

determine the largest set D of real numbers for which f : D => R is a function:

$\displaystyle h(x) = \sqrt{\frac{1}{x-6}+1}$

any ideas on how to do this practice question? cheers.

2. Originally Posted by jvignacio
determine the largest set D of real numbers for which f : D => R is a function:

$\displaystyle h(x) = \sqrt{\frac{1}{x-6}+1}$
any ideas on how to do this practice question? cheers.
X should be in the set D (called Domain)
Thus all x which give you a real y are in this domain as co-domain is R
Solve this inequality

$\displaystyle \frac{1}{x-6}+1 \ge 0$......so that we don't have negative inside root

And obviously $\displaystyle x \ne 6$

...since 0 in denominator is not allowed
Thus every x which satisfies the above two conditions is your answer

X should be in the set D (called Domain)
Thus all x which give you a real y are in this domain as co-domain is R
Solve this inequality

$\displaystyle \frac{1}{x-6}+1 \ge 0$......so that we don't have negative inside root

And obviously $\displaystyle x \ne 6$

...since 0 in denominator is not allowed
Thus every x which satisfies the above two conditions is your answer
thanks man! appreciated

4. is this true for

$\displaystyle \frac{1}{x-6}+1 \ge 0 = $$\displaystyle x \ge 5 ?? 5. Originally Posted by jvignacio is this true for \displaystyle \frac{1}{x-6}+1 \ge 0 =$$\displaystyle x \ge 5$

??
DO you think its correct for 5.5

Do you think its incorrect for 4

Answer is (-infinity , 5) And (6, infinity)

6. I think you are still in doubt , isn't it , I saw you peeking at this again and again

$\displaystyle \frac{1}{x-6} +1 \ge 0$

$\displaystyle \frac{x-5}{x-6} \ge 0$

---Now if $\displaystyle x> 6$the denominator on LHS is positive

Hence $\displaystyle x\ge 5$ but in the initial condition we have taken it greater than 6

Hence this is correct for $\displaystyle x>6$

---Now when $\displaystyle 5< x<6$ the denominator on LHS is negative

So there will be a change in inequality sign when we will multiply it by
(x-6) on both sides

Hence $\displaystyle x-5 < 0$

$\displaystyle \implies x < 5$ but this is against initial condtion hence no answer in this range of x

----When $\displaystyle x \le 5$ the denominator on LHS is negative
So on multiplying the sign of inequality changes

Hence $\displaystyle x-5 \le 0$

thus $\displaystyle x \le 5$ is an answer

Now for the complete answer we have to take a combination of all these solution sets(answer as in my last post)

------------------------------------

I am not sure if you were unclear about it, but I think if you were this will be helpful ,ask incase if you still find any trouble (remember the change of inequality sign on multiplying with a negative number on both sides)

I think you are still in doubt , isn't it , I saw you peeking at this again and again

$\displaystyle \frac{1}{x-6} +1 \ge 0$

$\displaystyle \frac{x-5}{x-6} \ge 0$

---Now if $\displaystyle x> 6$the denominator on LHS is positive

Hence $\displaystyle x\ge 5$ but in the initial condition we have taken it greater than 6

Hence this is correct for $\displaystyle x>6$

---Now when $\displaystyle 5< x<6$ the denominator on LHS is negative

So there will be a change in inequality sign when we will multiply it by
(x-6) on both sides

Hence $\displaystyle x-5 < 0$

$\displaystyle \implies x < 5$ but this is against initial condtion hence no answer in this range of x

----When $\displaystyle x \le 5$ the denominator on LHS is negative
So on multiplying the sign of inequality changes

Hence $\displaystyle x-5 \le 0$

thus $\displaystyle x \le 5$ is an answer

Now for the complete answer we have to take a combination of all these solution sets(answer as in my last post)

------------------------------------

I am not sure if you were unclear about it, but I think if you were this will be helpful ,ask incase if you still find any trouble (remember the change of inequality sign on multiplying with a negative number on both sides)

haha thanks for that, yeah ive been trying to study it and really try understand it. cheers for the last post. made things more simple