Which of the following pairs of functions are inverse functions?

Question

a. f(x)=3x-9, g(x)=-3x+9
b. f(x)=4-x, g(x)=4+x
c. f(x)=x+5, g(x)=x-5
d. f(x)=x, g(x)= -x

Answer 1

Note :

Two functions f and g are inverse functions if and only if both of their compositions are the identity function.

That means [f o g](x) = x and [g o f](x) = x.

a).f(x) = 3x - 9, g(x) = - 3x + 9.

Check to see if the compositions of f(x) and g(x) are identity functions.

[f o g](x) = f(g(x))                                            [g o f](x) = g(f(x))

              = f(- 3x + 9)                                                    = g(3x - 9)

              = 3(- 3x + 9) - 9                                             = - 3(3x - 9) + 9

              = - 9x + 27 - 9                                                = - 9x + 27 + 9

              = - 9x + 18.                                                    = - 9x + 36.

Therefore, [f o g](x) and [g o f](x) are not equal, so, the functions are not inverses.

b).f(x) = 4 - x, g(x) = 4 + x.

Check to see if the compositions of f(x) and g(x) are identity functions.

[f o g](x) = f(g(x))                                           [g o f](x) = g(f(x))

              = f(4 + x)                                                        = g(4 - x)

              = 4 - (4 + x)                                                    = 4 + (4 - x)

              = 4 - 4 - x                                                        = 4 + 4 - x

              = - x.                                                               = 8 - x.

Therefore, [f o g](x) and [g o f](x) are not equal, so, the functions are not inverses.

Answer 2

Contd....

c).f(x) =  x + 5, g(x) = x - 5.

Check to see if the compositions of f(x) and g(x) are identity functions.

[f o g](x) = f(g(x))                               [g o f](x) = g(f(x))

              = f(x - 5)                                             = g(x + 5)

              = (x - 5) + 5                                        = (x + 5) - 5

              = x - 5 + 5                                           = x + 5 - 5

              =  x.                                                    =  x.

Therefore, [f o g](x) and [g o f](x) equal to x, so, the functions are inverses.

d).f(x) =  x , g(x) = - x.

Check to see if the compositions of f(x) and g(x) are identity functions.

[f o g](x) = f(g(x))                            [g o f](x) = g(f(x))

              = f(- x)                                            = g(x)

              = - x.                                               = - x.

Therefore, [f o g](x) and [g o f](x) equal to - x, so, the functions are not inverses.