Find the inverse function of each of the following functions

Find the inverse function of each of the following functions. Specify the domains of the inverse functions. (a) f(x) = (1/5x) + 8

(b) h(x) = 5/(x + 3)

(c) g(x) = 43 x 7

(d) j(x) = x + x 1

e) k(x) = 16 x^2, 0 x 4

Solution

a) . f(x) = (1/5x) + 8

domain : 1/5x is not equal to zero

so x is not equal to zero

range : (-infinte , infinte)

inverse function:

let f(x) = y

x = f inverse (y)

y = (1/5x) + 8

y - 8 = (1/5x)

5x = 1 / y -8

x = (1/5(y-8) )

so f inverse of x = 1/5(x-8)

b) .

h(x) = 5/(x+3)

domain x+3 !=0

x != -3

x is not equal to -3

range : all the values

inverse function:

h(x) =y

y = 5 /(x+3)

(x+3) = 5/y

x = 5/y -3

h inverse of (x) = 5/x -3

c) .

g(x) = 43 x 7

domain: all values

range :alll values

inverse :

g(x) =y

y =43 x 7

y+7 - 43 = -x

x = -y -7 +43

so g inverse(x) = -x -7 +43

d)

j(x) = x + x 1

domain:

all the positve values

range : (-1 , infinte)

inverse:

j(x) = x + x 1

j(X) =y

y = 2x -1

2x = y+1

x = (y+1)/2

x = [ (y+1) /2]^2

so j inverse (x) = [ (x+1) /2]^2

e)

k(x) = 16 x^2, 0 x 4

domain: 0 <= x <4

range: 0 x 4

inverse:

k(x) =y

16 x^2 =y

16 -x^2 =y^2

-x^2 =y^2 -16

x^2 =16-y^2

x = sqrt(16-y^2)

k inverse (x) =16 x^2