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

Question

Elijah Abram · Accepted Answer

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:

Elijah Abram · Answer

undefined To sketch the curve of the function $ f(x) = x + \sqrt{1 - x} $:

1. Identify the domain of the function, which is $ x \leq 1 $ due to the square root term.
2. Determine the behavior of the function as $ x $ approaches the domain boundaries, i.e., $ x \to -\infty $ and $ x \to 1 $.
3. Find critical points by setting the derivative of the function equal to zero and solving for $ x $.
4. Determine the behavior of the function around critical points using the first derivative test.
5. Identify any vertical asymptotes by checking for values of $ x $ that make the denominator of the function zero.
6. Plot key points, such as intercepts, critical points, and asymptotes.
7. Sketch the curve, ensuring it adheres to the identified characteristics and behaviors.