Is the given information in this inverse function problem incorrect?

Question

y = x^2 + 4x + 10

(x ≥ 0)

What is the inverse function for this function? Also, what are the domain and range of the inverse?


I've already worked this problem, but once I have the inverse, I get the wrong range. I understand that the range of the inverse function is the domain of the original function, which is given in the problem. But instead of (x ≥ 0), I get (y ≥ - 2) (y because it's the range of the inverse, but that would be (x ≥ - 2) if I were saying it's the domain of the original). Is the domain I was given incorrect?

When I graph the original function, I can see that the vertex of the parabola (which opens up) is at x = - 2 , which also leads me to believe that I was given an incorrect domain for my original function.

Am I misunderstanding something or is (x ≥ - 2) what the domain for the original function should've been?

By the way, I got my inverse function to be y = (-4+√(4x-24)) / 2

Thank you! 10 points for the best explanation!

Answer 1

The domain can't be "incorrect". You should take that as part of the definition of this function. It's the function that takes values x >= 0 and maps them to x^2 + 4x + 10. It's a portion of a parabola, not the whole thing. Specifically, the portion of the parabola to the right of x = 0. Graphically, the inverse function will be that portion of the parabola turned sideways.



Yes, the vertex is at x = -2. So you know that this portion of the parabola is invertible, since it's only a part of the graph to the right of that vertex. You won't have two possible x values for a given y value.



It could just as easily have defined the original domain to be x >= 100, meaning your function is only defined on those values of x >= 100.



The range of your original function is the set of y values that occur when x >= 0. So for instance y = 6 is not in the range because that value occurs when x = -2, but does not occur for any x in the defined domain.



The range of your inverse function better end up being x >= 0. To make sure that happens, the domain is the set of y values that produce values of x >= 0. Since the original function has the value of y = 10 at x = 0 and it increases monotonically for x > 0, then the range of the original function is y >= 0, and that will be the domain of the inverse function.

Answer 2

The domain is stated as x >= 0 and you can't change that.

Notice that x = 0 ----> y = 10 so the range of the original function is y >= 10



You have the right inverse function but it can be simplified to

y = -2 + sqrt(x - 6).

This will be positive because x >= 10.



The domain and range of the inverse function are x >= 10, y >= 0. As you suspected they are the range and domain of the original function swopped over.



It may be helpful to remember that the graph of the inverse of a function is the original graph reflected in the line y = x. The original graph started at (0, 10) and the inverse graph starts at (10, 0).

Answer 3

You are correct in that the given function is 1 to 1 in the domain (x>= -2). The specified domain is a subset of that domain so it's not an error. They simply have decided not to consider any x<0. For this reason the range of the inverse is y>=0.