## anonymous one year ago On combining functions, and composite functions. How do I deriving, the domain, and range of such functions. An example: $f(x) = \sqrt{x - 1} \\ g(x) = 3x + 1$ $P(x) = \left(\dfrac{b}{g}\right)(x)$

1. anonymous

I have so far: $P(x) = \left(\dfrac{\sqrt{x - 1}}{3x + 1}\right)(x)$ I also know the deniominator $$\ne$$ $$0$$.

2. anonymous

I just noticed the numerator cannot be 1, or smaller as that would cause imaginary numbers.

3. anonymous

Wait, it can be $$1$$ since that just makes the numerator zero.

4. zzr0ck3r

you need the following to be true $$x-1\ge 0$$ and $$3x+1\ne 0$$ For the first you get $$x\ge 1$$ and for the second you get $$x>\frac{-1}{3}$$ The $$x\ge 1$$ trumps the $$x\ge \frac{_1}{3}$$ so the domain is $\{x\mid x\ge 1\}$

5. zzr0ck3r

that should say $$x\ne \frac{-1}{3}$$ not $$x\ge \frac{1}{3}$$

6. anonymous

All right, thank you. (Sorry for the late reply, I was away from my keyboard.) Sub question: Would $$\{x \in ℝ\mid x\ge 1\}$$ be the same as you wrote? (That is the notation use in my class on a question like this, so I wonder if it is just an optional notation, or has a significant meaning in this case. (I understand that is means x is all reals, just that does your notation already imply that without the usage of it, or am I using it wrong in this case.))

7. zzr0ck3r

Nah, I was just assuming \(\mathbb{R}) was the universal set, in which case it is implied. But I should be more accurate.