Question 285588
(f+g)(x) = f(x) + g(x)


f(x) = 2x-5
g(x) = sqrt(x+6)


(f+g)(x) = (2x-5) + sqrt(x+6)


The only restriction on the domain of this function is the square root of (x+2^ can't be negative.


to find out when sqrt(x+6) goes negative, use the equation:


(x+6) < 0


Subtract x from this equation to get x < -6


When x < -6, the square root of (x+6) goes negative and the domain goes invalid.


The domain of (f+g)(x) is therefore all values of x >= -6


Note that this i also the domain of g(x).


The domain of f(x) is all real value of x.


(f+g)(x) adds f(x) plus g(x) together so the domain becomes the more restrictive of the two which is the domain of g(x).


A graph of the equation (f+g)(x) = (2x-5) + sqrt(x+6) is shown below:


{{{graph (400,400,-10,10,-20,20,(2x-5) + sqrt(x+6))}}}


You can see that the graph stops at x = -6 and doesn't go any further.


That's because the value of y when x < -6 is not a real number and is therefore undefined.