1. ## Implied Domain

How do I work out the implied domains for $\displaystyle f(x)=\frac{x^2-1}{x+1}$ and also for $\displaystyle f(x)=\sqrt\frac{x-1}{x+2}$

Thanks heaps

2. Originally Posted by Stroodle
How do I work out the implied domains for $\displaystyle f(x)=\frac{x^2-1}{x+1}$ and also for $\displaystyle f(x)=\sqrt\frac{x-1}{x+2}$

Thanks heaps
For the first one you require all values of $\displaystyle x$ such that $\displaystyle x + 1 \neq 0$.

For the second one, you require all values of $\displaystyle x$ such that $\displaystyle \frac{x - 1}{x + 2} \geq 0$.

3. Thanks. I get the first one now, but my problem is solving $\displaystyle \frac{x-1}{x+2}\geq0$

I'm getting $\displaystyle 1-\frac{3}{x+2}$ so is the domain R\{-2} ?

What's the command for squiggly brackets in latex btw?

4. I think for the first one there are no restrictions, because $\displaystyle \frac{x^2-1}{x+1}=x-1$ so the implied domain should be $\displaystyle \mathbb{R}$.

5. For the first one I'm getting $\displaystyle \frac{x^2-1}{x+1}=2+\frac{-2}{x+1}$

6. Originally Posted by Stroodle
For the first one I'm getting $\displaystyle \frac{x^2-1}{x+1}=2+\frac{-2}{x+1}$
That's not correct.
$\displaystyle 2+\frac{-2}{x+1}=\frac{2x}{x+1}$

7. hi
for the first one : $\displaystyle \mathbb{R}$
for the second : $\displaystyle ]-\infty,-2[ \cup [1,+\infty[$

8. Hmm. I'm definitely doing something wrong here. I'm getting $\displaystyle \frac{x^2-1}{x+1}=\frac{(x+1)(x-1)}{x+1}=1+\frac{x-1}{x+1}=1+\frac{(x+1)-2}{x+1}=2+\frac{-2}{x+1}$
Which must be wrong. But the answer is definitely R/{-1}

9. Originally Posted by Raoh
hi
for the first one : $\displaystyle \mathbb{R}$
for the second : $\displaystyle ]-\infty,-2[ \cup [1,+\infty[$
Thanks Raoh.
How'd you get the second answer?

10. Originally Posted by AMI
I think for the first one there are no restrictions, because $\displaystyle \frac{x^2-1}{x+1}=x-1$ so the implied domain should be $\displaystyle \mathbb{R}$.
Incorrect. If $\displaystyle x = -1$ then you have just committed the cardinal sin of dividing by zero when you do this. The domain is all real numbers EXCEPT $\displaystyle x = -1$. The graph of $\displaystyle y = \frac{x^2 - 1}{x + 1}$ has a 'hole' at $\displaystyle x = -1$.

Originally Posted by Raoh
hi
for the first one : $\displaystyle \mathbb{R}$
[snip]
Incorrect. See above.

Originally Posted by Raoh
[snip]
for the second : $\displaystyle ]-\infty,-2[ \cup [1,+\infty[$
Your bracket notation needs some minor corrections:

$\displaystyle {\color{red} (}-\infty,-2{\color{red})} \cup [1,+\infty{\color{red})}$

11. Originally Posted by Stroodle
Hmm. I'm definitely doing something wrong here. I'm getting $\displaystyle \frac{x^2-1}{x+1}=\frac{(x+1)(x-1)}{x+1}$ Mr F says: Correct. Now cancel the common factor of (x + 1) to get x - 1. You can do this provided $\displaystyle {\color{red}x \neq -1}$ (otherwise you're dividing by zero).

$\displaystyle =1+\frac{x-1}{x+1}=1+\frac{(x+1)-2}{x+1}=2+\frac{-2}{x+1}$ Mr F says: Wrong.
Which must be wrong. But the answer is definitely R/{-1}
Originally Posted by Stroodle
Originally Posted by Raoh
hi

for the first one : $\displaystyle \mathbb{R}$

for the second : $\displaystyle ]-\infty,-2[ \cup [1,+\infty[$
Thanks Raoh.

How'd you get the second answer?
Originally Posted by mr fantastic
[snip]
For the second one, you require all values of $\displaystyle x$ such that $\displaystyle \frac{x - 1}{x + 2} \geq 0$.
Case 1: $\displaystyle x - 1 \geq 0$ AND $\displaystyle x + 2 > 0 \Rightarrow x \geq 1$.

Case 2: $\displaystyle x - 1 \leq 0$ AND $\displaystyle x + 2 < 0 \Rightarrow x < -2$.

12. $\displaystyle \mathbb{R}/\left \{ -1 \right \}$ (sorry about later)
for the second one : study the sign of $\displaystyle x-1$ and $\displaystyle x+2$ then study $\displaystyle \frac{x-1}{x+2}$
and see where $\displaystyle \frac{x-1}{x+2}$ is positive.

13. Awesome. Thanks. I made some pretty silly errors there

14. Originally Posted by mr fantastic
Incorrect. If $\displaystyle x = -1$ then you have just committed the cardinal sin of dividing by zero when you do this. The domain is all real numbers EXCEPT $\displaystyle x = -1$. The graph of $\displaystyle y = \frac{x^2 - 1}{x + 1}$ has a 'hole' at $\displaystyle x = -1$.
My judgement was that we don't actually divide by $\displaystyle x+1$ since the final expression is $\displaystyle x-1$, but I understand what you mean.
So, for example, $\displaystyle \frac{x^2}{x}$ will not be defined in $\displaystyle 0$ even though it's equal to $\displaystyle x$??

15. Oh, nevermind, I think I got it.
I think I was saying something like "we don't need any restrictions for the function $\displaystyle x\mapsto (\sqrt{x})^2$ because it's equal to $\displaystyle x$", which is obviously false