Answer:
The second option is the correct answer.
Step-by-step explanation:
The range of a function of variable x means the set of all possible values of the function.
The function, is given by [tex]f(x) = -1 -x, x < 0.\\f(x) = 1 + \sqrt{x} , x \geq 0.[/tex].
[tex]\frac{d f(x)}{dx} = -1, x < 0.\\\frac{d f(x)}{dx} = \frac{1}{2\sqrt{x} } , x \geq 0.[/tex].
For x < 0, we can not find any maximum or minimum value. So, in this case we need to find the limits.
[tex]\lim_{x \to -\infty} (-1 - x) = \infty.[/tex].
Here, if you notice you will see as the function's value is - 1 - x, for x<0, the ultimate value of the function for all x<0 will be greater than - 1.
So, for x < 0 all the values of the function will be > -1.
For x > 0, [tex]\lim_{x \to \infty} (1 + \sqrt{x} ) = \infty[/tex].
For x = 0, the function's value will be 0.
The range of the function will be the all real numbers y > -1.
