The one-to-one functions and are defined as follows.?

Question

 

g={(-8,-2),(-1,7),(3,-1),(9,3)}
h(x)=3x+2

find
g^-1(-1)=?
h^-1(x)=?
(h o h^-1) (8)=

 

Answer 1

ii) Given function h(x) = 3x+2.

let h(x) = y then x = h^-1(y)

y = 3x+2

 

Siubtract 2 from each side

y - 2 = 3x + 2 - 2

 

Additive inverse property: 2 - 2 = 0

y - 2 = 3x

 

Divide each side by 3

(y - 2)/3 = 3x/3

 

Cancel common terms

x = (y - 2)/3

 

Then h^-1(y) = (y - 2)/3

          h^-1(x) = (x - 2)/3

 

iii) (h o h^-1) (8) = h(h^-1(8))

h^-1(x) = (x - 2)/3 then h^-1(8) = (8 - 2)/3

                                                = 6/3

                                                = 2

 

Substitute h^-1(8) = 2 in (h o h^-1) (8)  then

h(x) = 3x+2

 

Substitute x = 2

h(2) = 3(2)+2

        = 6+2

        = 8

Answer 2

i) g = {(-8,-2),(-1,7),(3,-1),(9,3)} which means that g(-8) = -2, g(-1) = 7, g(3) = -1 and g(9) = 3.

Therefore g^-1 = {(-2,-8),(7,-1),(-1,3),(3,9)} which means that g^-1(-2) = -8, g^-1(7) = -1, g^-1(-1) = 3 and g^-1(3) = 9.

We have to find g^-1(-1) which is 3.

Therefore g^-1(-1) = 3.