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

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

Lastly, we find:

$\begin{gathered} (h.h^{-1})(x)=\frac{x+13}{4}\times4x-13 \\ (h.h^{-1})(1)=\frac{1+13}{4}\times4(1)-13 \\ (h.h^{-1})(1)=\frac{14}{4}(4-13) \\ (h.h^{-1})(1)=\frac{14(-9)}{4}=-\frac{63}{2}=-31.5_{} \end{gathered}$