College Algebra Questions

Please help me :)

A stone is dropped into a pond, causing a circular ripple that is expanding at a rate of 10 ft/sec. Describe the area of the circle, A , as a function of time, t .

Enter an exact answer in terms of pie.

Check whether f and g are inverses of each other.

f(x) = 2x + 2 and x-2/2

(f*g)(x)
(g*f)(x)

Find the inverse for the given function.

5/x+7

Hi Mandy,

(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;