(1) The key thing you need to know in this is that when the ripple is expanding, that it is the radius of the circle that is increasing. Thus, as a function of time, the radius, r(t) = 10t, assuming we're starting at a radius = 0 at t=0.
Now just apply the area formula:
A = pi r^2 = pi * (10t)^2 = pi * 100 t^2 = 100pi t^2.
(2) For your second problem, to determine whether functions are inverses of each other you have to show that f*g = g*f = x. That is, no matter which function you start with, by composing it with the other, you back the original input.
In your case, you f(x) = 2x + 2 and g(x) = (x-2)/2 [I'll assume you meant to group the x-2 in the g(x), as it would be kind of silly to start off with an expression that is so easily reduceable.]
f*g(x) = f(g(x)) = f([(x-2)/2]) = 2*[(x-2)/2] + 2 = (x-2) + 2 = x. CHECK!
g*f(x) = g(f(x)) = g(2x+2) = [(2x+2)-2]/2 = 2x/2 = x. CHECK! Thus, f and g are inverses.
(3) To find the inverse of a given function in x, simply set it equal to y. Then swap x and y, and solve for y:
(a) In case you intended to write the problem just as you typed it: y = 5/x + 7; then the inverse satisfies the relation: x = 5/y + 7. Now make it explicit in terms of y:
x = 5/y + 7;
x - 7 = 5/y
y(x-7) = 5
y = 5/(x-7).
(b) Just in case you meant to type: y = 5/(x+7); then the inverse satisfies the relation: x = 5/(y+7). Now make it explicit in terms of y:
x = 5/(y+7);
x(y+7) = 5;
y + 7 = 5/x
y = 5/x - 7.
Both (a) and (b) can be checked by using the method in (2) above.
To help reinforce your understanding of inverse functions, I've searched and found a webpage and a video tutorial that address problems similar to this one, and I thought they might be helpful to you. I've listed them below.
As always, if you need more help, please clarify where you are in the process and what's giving you trouble. I'd be more than happy to continue to assist.
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