Hello everyone! I just need help with two of these questions on my math homework.
The first question is:
For b > 0, what are the domain and range of b^x?
The second question is:
Please explain it as well because I would actually like to understand how the problems were solved as opposed to just being given an answer.
Oh, and for the second problem, I have to express my answer in terms of a, h, and k, not solve for them. Thats what confused me; the fact no numbers were involved [in expressing the sides].
For example, on the left triangle, sin75 = h/a, cos15 = h/a.
On the right triangle, sin45 = k/a and cos45 = a/k. That would make the tangent = (k/a) / (a/k), I'm not sure if I can simplify it further (maybe [k/a]^2). That gives me the base for the right triangle. I'm still totally lost on the left triangle though.
However, what would I do with that information? Since its expressed in letters, how would I solve for the side that does not have a letter assigned to it?
Try using x = -infinity, x = 0 and x = infinity. Those three points often help in determining the domain.
By the way, do you know the shape of the graph ? This might help you.
2. For this one, I think there is a simpler way, that is using Pythagoras' Theorem. Do you know about it?
The domain seems to be all real numbers, or negative infinity to positive infinity.
Yeah, I know the pythagorean theorem. I thought of that at first but I don't know how I would express the answer if one of the sides on both triangles is unidentified.2. For this one, I think there is a simpler way, that is using Pythagoras' Theorem. Do you know about it?
In both cases though, x is not identified/given a letter, so I don't know how to express the answer in terms of only h, a, and k.
Would I do and ?
Would the answer result in
1. Yes, the domain is negative infinity to infinity.
The range, however is y > 0. The graph never touches the x-axis.
2. Be careful!
Let y be the longer length, and x be the shorter length.
x + y is what you're looking for.
That's not too easy on the eyes though. Let's take trigonometry.
I got the width of the alley in terms of a, h and k first using Pythagoras' Theorem.
Then, I got the width of the alley by using trigonometry, but then, it's only in terms of a.
Those are two different methods.
I could have made use of other trigonometric ratios.
For example tan.
Alley width =
Alley width =