Question

How do you sketch the curve `f(x)=x+sqrt(1-x)` ?

Answer 1

First I would check the square root. What I want to avoid is to have a negative argument. This is because I cannot find a Real Number that is solution of a negative square root.
So I say that:
`1-x` must be `>0`
Let us see what this condition tells us about the "permitted" values of `x` for our function:
`1-x>0`
`-x> -1`
and finally:
`x<1`
This means that I can choose only values in the interval between `1` and `-oo`.

I then try to use values of `x` starting from 1 and going towards -1 to see a possible tendency of my curve.

I then test what is going to happen when `x->-oo`.

Choosing `x` very big negatively I have in my function a situation like this:
`f(-1,000,000)=-1,000,000+sqrt(1,000,001)=(-1,000,000+1000)=-999000`
This is to say that `-oo` will always win and even if in your function you have a positive part (`sqrt(1-x`) it does not interfere too much with the `-oo` tendency of the complete function.

Finally, the graph should look like this: