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.
Thats exactly how the problem was phrased, though. There was no additional information included. I'll even snapshot it so that you can make sure.
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?
1.
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?
I just googled the graph to make sure, and yes I've seen it before. Based on it, it looks like the range is since it never falls below zero on the left side and shoots up infinitely on the right.
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.
Then;
x + y is what you're looking for.
So,
That's not too easy on the eyes though. Let's take trigonometry.
So;
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 =
With sin.
Alley width =