Question

the one to one function g and h are defined as follows g={(-1,8),(1,3),(6,4),(8,5),(9,-9)}h(x)=x-8/7Find the following. g^-1 (8) = h^-1 (x) =(h • h^-1)(o) =

Answer 1

One to one function : if no two elements in the domain of f correspond to the same element in the range of f .

The function g and h are one to one function and defined as;

g={(-1,8),(1,3),(6,4),(8,5),(9,-9)}

$h(x)=\frac{x-8}{7}$

1)

$\begin{gathered} g^{-1}(8) \\ \text{From the defination of function g;} \\ g=\text{ (-1,8)} \\ i\mathrm{}e\text{. it defined as g(-1) = 8} \\ \text{thus, -1=g}^{-1}(8) \\ g^{-1}(8)=-1 \end{gathered}$

2)

$\begin{gathered} h^{-1}(x) \\ \text{The function h(x) defined as;} \\ h(x)=\frac{x-8}{7} \\ \text{Let h(x)=y} \\ y=\frac{x-8}{7} \\ \text{Solve for x;} \\ 7y=x-8 \\ x=7y+8 \\ h^{-1}(x)=7x+8 \end{gathered}$

3)

$\begin{gathered} (h\circ h^{-1})(0) \\ (h\circ h^{-1})(x)=h(h^{-1}(x)) \\ (h\circ h^{-1})(x)=h(7x+8) \\ (h\circ h^{-1})(x)=\frac{(7x+8)-8}{7} \\ (h\circ h^{-1})(x)=\frac{7x+8-8}{7} \\ (h\circ h^{-1})(x)=\frac{7x+0}{7} \\ (h\circ h^{-1})(x)=x \\ \text{Substitute x = 0} \\ (h\circ h^{-1})(0)=0 \end{gathered}$

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