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

Dec 26, 2014

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 $- \infty$.

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 $x \to - \infty$.

Choosing $x$ very big negatively I have in my function a situation like this:
$f \left(- 1 , 000 , 000\right) = - 1 , 000 , 000 + \sqrt{1 , 000 , 001} = \left(- 1 , 000 , 000 + 1000\right) = - 999000$
This is to say that $- \infty$ 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 $- \infty$ tendency of the complete function.

Finally, the graph should look like this: 