Question

Which of the following are characteristics of the graph of the square root

Answer 1

Choices A and B have typos in them, so its not clear what you're trying to say for those parts. However, the domain of $y = \sqrt{x}$ is $x \ge 0$ meaning that x can be 0 or larger. In other words, we can't have x be negative. Similarly, y is the same story because $y = \sqrt{x}$ has the inverse $y = x^2$, but only when $x \ge 0$, so therefore $y \ge 0$ as well. In short you can say both x and y are nonnegative.

To summarize so far, the domain is $x \ge 0$ and the range is $y \ge 0$

Since x = 0 and y = 0 are the smallest x and y values possible, this means (x,y) = (0,0) is the left-most point or where the graph starts. This is the origin. Choice C is a true statement.

Choice D on the other hand is not a true statement. Graph out $y = \sqrt{x}$ and you'll see that a straight line does not form, but instead a nonlinear curve that grows forever. That growth gradually diminishes as x gets larger. Algebraically you can pick three points from the function and show that the slopes are different. Say the three points are P, Q and R. If you can show that slope of PQ does not equal slope of QR, then the function is not linear.