FINDING AN INVERSE FUNCTION VALUE GIVEN A FORMULA

Problem 1 :

Given h(x) = 5 - 9x find h^-1(x), evaluate f^-1(-3).

Solution :

h(x) = 5 - 9x

Let h(x) = y

y = 5 - 9x

Interchange x and y.

x = 5 - 9y

Solve for y.

9y = 5 - x

y = (5 - x)/9

Replace y by h^-1(x) = (5 - x)/9.

f^-1(-3) = (5 - (-3))/9

= (5 + 3)/9

f^-1(-3) = 8/9

Problem 2 :

Given g(x) = (1/2)x + 7 find g^-1(x), evaluate f^-1(1).

Solution :

g(x) = (1/2)x + 7

Let g(x) = y

y = (1/2)x + 7

Interchange x and y.

x = (1/2)y + 7

Solve for y.

2x = y + 14

y = 2x - 14

Replace y by g^-1(x) = 2x - 14

f^-1(1) = 2(1) - 14

= 2 - 14

f^-1(1) = -12

Problem 3 :

Given f(x) = (x - 2)³ + 1 find f^-1(x), evaluate f^-1(5).

Solution :

f(x) = (x - 2)³ + 1

Let f(x) = y

y = (x - 2)³ + 1

Interchange x and y.

x = (y - 2)³ + 1

Solve for y.

(y - 2)³ + 1 = x

(y - 2)³ = x - 1

Take cube root on both sides,

y - 2 = ∛(x - 1)

y = 2 + ∛(x - 1)

Replace y by g^-1(x) = 2 + ∛(x - 1)

f^-1(5) = 2 + ∛(x - 1)

= 2 + ∛(5 - 1)

= 2 + ∛4

Problem 4:

Given, A(x) = y find A^-1(x).

Solution:

A(x) =

Let A(x) = y

y =

Interchange x and y.

x =

Solve for y.

2y + 11 = x⁵

2y = x⁵ - 11

y = (x⁵ - 11)/2

Replace y by A^-1(x) = (x⁵ - 11)/2

Problem 5:

Given f(x) = 4x/5 - x find f^-1(x).

Solution :

f(x) = 4x/5 - x

Let f(x) = y

y = 4x/5 - x

Interchange x and y.

x = 4y/5 - y            

Solve for y.   

x(5 - y) = 4y

5x - xy = 4y

5x = 4y + xy

5x = y(4 + x)

y = 5x/4 + x

Replace y by f^-1(x) = 5x/4 + x.

Problem 6 :

Given h(x) = (1 + 2x) / (7 + x) find h^-1(x).

Solution :

h(x) = (1 + 2x) / (7 + x)

Let h(x) = y

y = (1 + 2x) / (7 + x)

Interchange x and y.

x = (1 + 2y) / (7 + y)

Solve for y.

x(7 + y) = 1 + 2y

7x + xy = 1 + 2y

xy - 2y = 1 - 7x

y(x - 2) = 1 - 7x

y = (1 - 7x) / (x - 2)

Replace y by h^-1(x) = (1 - 7x)/(x - 2).