help me with these stupid inverse functions?

Question

how do i find a formula for the inverse function these 1-1 functions. Also the range and domain

1. f(x) = ½x - 4

2. g(x) = (2x-7)/(x-4)

3. h(x) = x-1 if 0 ≤ x < 1
= 2 - x if 1≤ x ≤ 2

4. F(x) = sin(x) , x is in ( π/2 , 3π/2 )

Answer 1

Ok, let's go on :

To find the inverse function you need, to do this :

Note : Domain of the function = Range of the inverse function

Range of the function = Domain of the inverse function

1. f(x) = ½x - 4

domain : Reals

range : Reals

y = 1/2*x - 4

x = 1/2*y - 4

then :

x + 4 = 1/2*y

y = 2x + 8 >>> inverse function

domain : Reals

range : reals

2. g(x) = (2x-7)/(x-4)

domain : Reals - [ 4]

range : Reals - [2]

The same : y = 2x -7 / x-4

x = 2y - 7 / y - 4

x = 2 + (1 / y - 4 )

x-2 = 1 / y-4

y-4 = 1 / x-2

y = 4 + (1 / x-2) >>> inverse function

Domain : Reals - [2]

Range : Reals - [4]

3. h(x) = x-1 if 0 ≤ x < 1and = 2 - x if 1≤ x ≤ 2

if 0 ≤ x < 1

y = x - 1

x = y - 1

y = x+1 >>> inverse function for x E [ 0 ; 1 ]

if 1≤ x ≤ 2

y = 2 - x

x = 2 - y

y = 2 - x >>> inverse function

4. F(x) = sin(x) , x is in ( π/2 , 3π/2 )

range of F(x) = < -1 ; 1 >

y = sin(x)

x = sin(y)

y = arcsin(x)

the domain of this function will be the range of the function : sin(x)

y = arcsin(x)

x E < -1 ; 1 >

Hope that helped

Answer 2

definition of inverse function: swap y and x, and solve for y
I will only do the first two.

1. f(x) = y = 1/2 x - 4
x = 1/2 y - 4
x +4 = 1/2 y (multiply both sides by 2)
2x +8 = y (this is the inverse)
domain: all real numbers
range: all real numbers
why? because, it is a linear function and goes to infinity (left and right) so no restrictions

2. g(x) = (2x-7)/(x-4) = y , x cannot be = 4 because its dividing by 0 which is undefined

(2y-7)/(y-4) = x (swap x and y)
2y-7 = x(y-4) (multiply x-4 on both sides)
2y-7 = xy - 4x (expand)
2y - xy = 7 - 4x (move y's to one side)
y (2-x ) = 7 - 4x (factor y out)
y = (7-4x)/(2-x) (isolate y which is multiply both sides by (2-x)

this one's domain and range are more difficult to find
hmm

it is a variation of y = 1/x type. asymptote is 2 because
otherwise it is dividing by 0. simply find it by finding x=0 in the
denominator. after graphing it on TI-83+,
domain: all real numbers but x cannot be = 2, range: all real numbers

hope that helps

Answer 3

I consider it makes extra feel the school approach. I role is a mapping so f(x) maps x ----> f(x) f(x)^-one million maps f(x) ----> x What is the argument of f(x)^-one million? It is f(x) so f(f(x))^one million = x. (You're going from x to f(x) after which from f(x) to x) Does consider make feel?