MATH - f INVERSE functions?

Question

1. f(x)= .5x-10
between
-2 less than or equal to x
and 6 is greater than or equal to x
Find the domain of f inversve

2. f(x)=2x^3+4x+2
let g be the inverse function
g(x)= finverse (x)
find first deriv of g(-4)

Answer 1

1. What you want to do is replace your x with f(x) and f(x) with x and solve for f(x) this will give your inverse function.

f(x)= .5x-10
Replace x and f(x)
x = .5f(x) - 10
now solve for f(x), First we add 10 to both sides
x + 10 = .5f(x)
now we divide both sides by .5, which is equivalent to multiplying both sides by 2 because .5 = 1/2, so we get
2(x + 10) = f(x)
distribute the 2
2x + 20 = f(x) this is your inverse function. Now you domain and range simply swap. That is your domain, (from the original) becomes range (for your inverse) and your range (from the orignal) becomes doman (for your inverse). Since the range was not, given for the original, but it is easily to figure out. We are given the domain from which the function is limited [-2, 6], we must find the range of the function in that domain which is [-11, -7]. So your domain for the inverse function will be [-11, -7]

EDIT: The way i figured out your range for the original function is this. First i took -2 and plugged it back into the original function and then i took 6 and plugged it back into the equation. That is first I
f(-2) = .5(-2) - 10
= -1 -10
= -11
and then
f(6) = .5(6) - 10
= 3 - 10
= -7
therefore ur domain is -11 less than or equal to x and -7 greater than or equal to x. This then becomes your range for the inverse. I only did one... but if you need help with the other one IM ME or Email me. Goodluck

Answer 2

1. Since f is strictly increasing, its range is from f(-2) to f(6) inclusive, i.e. [-9, -7]. This is also the domain of the inverse.

2. Since g is the inverse of f we have g'(f(x)) = 1 / f'(x) [also written as g'(y) = 1 / f'(g(y)) - same thing since f(x) = y <=> x = g(y)]. For g'(-4) we need to find g(-4) first, i.e. find x such that
2x^3 + 4x + 2 = -4
=> 2x^3 + 4x + 6 = 0
(note x = -1 is a solution - better check there aren't any others!)
=> (x+1)(2x^2 - 2x + 6) = 0
The discriminant for the quadratic factor is (-2)^2 - 4(2)(6) = -44 so there are no further solutions. Hence x = -1.
f'(x) = 6x^2 + 4 = 10 at x = -1. So g'(-4) = 1/f'(-1) = 1/10.

Answer 3

f(x) = 0.5x - 10

-2 <= x <= 6

The domain of f inverse will be the range of f.
To find the range of f, all we have to do is apply f to the interval. That is, since

-2 <= x <= 6, it follows that
f(-2) <= f(x) <= f(6)

f(-2) = 0.5(-2) - 10 = -1 - 10 = -11
f(6) = 0.5(6) - 10 = 3 - 10 = -7

Therefore, -11 <= f(x) <= -7

That means the domain of f inverse will be -11 <= x <= -7.

Answer 4

So swap x for y and vice versa: x = .5y-10, resolve for y: y=2x+20, [-2,6], so now find the domain. Proceed the same way with 2, take the first derivative and plug in -4

Answer 5

When you have to find the inverse, substitute y for f(x) y= x/x-2 Now make x the subject of the formula. y(x-2) = x yx- 2y= x -2y= x- yx -2y= x(1-y) -2y/(1-y) =x Now substitute f'(x) for x and x for y Giving: f'(x) = -2x/1-x. This can be rearranged to give: 2x/x-1