[tex]\dfrac{4x + 5}{x - 5} = y \Rightarrow xy - 5y = 4x + 5 \Rightarrow xy - 4x = 5y + 5 \Rightarrow x = \dfrac{5y + 5}{y - 4}[/tex]
[tex]\Rightarrow \boldsymbol{g^{-1}(x) = \dfrac{5x + 5}{x - 4}}[/tex]
[tex]2x - 3 = y \Rightarrow 2x = y + 3 \Rightarrow x = \dfrac{y + 3}{2}[/tex]
[tex]\Rightarrow \boldsymbol{h^{-1}(x) = \dfrac{x + 3}{2}}[/tex]
[tex](g \circ h)(x) = \dfrac{4 \cdot h(x) + 5}{h(x) - 5} = \dfrac{4(2x - 3) + 5}{(2x - 3) - 5} = \dfrac{8x - 7}{2x - 8}[/tex]
[tex]\dfrac{8x - 7}{2x - 8} = y \Rightarrow 2xy - 8y = 8x - 7 \Rightarrow 2xy - 8x = 8y - 7 \Rightarrow x(2y - 8) = 8y - 7 \Rightarrow x = \dfrac{8y - 7}{2(y - 4)}[/tex]
[tex]\Rightarrow \boldsymbol{(g \circ h)^{-1}(x) = \dfrac{8x - 7}{2(x - 4)} }[/tex]
[tex](h \circ g)(x) = 2 \g(x) - 3 = 2 \cdot \dfrac{4x + 5}{x - 5} - 3 = \dfrac{2(4x + 5)-3(x - 5)}{x - 5} = \dfrac{5x + 25}{x - 5}[/tex]
[tex]\dfrac{5x + 25}{x - 5} = y \Rightarrow xy - 5y = 5x + 25 \Rightarrow xy - 5x = 5y + 25 \Rightarrow x(y - 5) = 5(y + 5) \Rightarrow x = \dfrac{5(y + 5)}{y - 5}[/tex]
[tex]\Rightarrow \boldsymbol{(h \circ g)^{-1}(x) = \dfrac{5(y + 5)}{y - 5} }[/tex]
