Question

The one-to-one functions g and h are defined as follows.g={(-9, -3), (2, -5), (3, -6), (5, 2)}h(x) = 4x -13Find the following.

Answer 1

We have been given the domain-range pairs of the function g.

We have a pair of domain-range pair as (5, 2)

Therefore,

`\begin{gathered} g^(-1)(2)=5 \\ \text{The }g^(-1)\text{ refers to the inverse and indicates that the domain and range places are } \\ interchanged. \\ \text{The range and domain of }g^(-1)\text{ is the domain and range of g respectively} \end{gathered}`

Next, we find

`\begin{gathered} h^(-1)(x)\text{ where h(x)=}4x-13 \\ We\text{ can call h(x) = y} \\ \text{therefore, y = }4x-13 \\ \text{Making x the subject of the formulae yields} \\ y+13=4x\text \\ x=(y+13)/(4)\text Dividing both sides by 4 \\ h^(-1)(x)=(y+13)/(4) \\ \text{Lastly, we interchange y with x} \\ h^(-1)(x)=(x+13)/(4) \end{gathered}`

Lastly, we find:

`\begin{gathered} (h.h^(-1))(x)=(x+13)/(4)*4x-13 \\ (h.h^(-1))(1)=(1+13)/(4)*4(1)-13 \\ (h.h^(-1))(1)=(14)/(4)(4-13) \\ (h.h^(-1))(1)=(14(-9))/(4)=-(63)/(2)=-31.5_{} \end{gathered}`