[tex]f,g:\mathbb_{R} \rightarrow \mathbb_{R}, $ $ $ $f(x) = 3x+5, $ $ $ $ g(x) = mx+m-2 \\ \\ 3x+5 = y \Rightarrow 3x = y-5 \Rightarrow x = \dfrac{y-5}{3} \Rightarrow x = \dfrac{y}{3}-\dfrac{5}{3}\Rightarrow \\ \\ \Rightarrow x = \dfrac{1}{3}\cdot y-\dfrac{5}{3} \Rightarrow f^{-1}(x) = \dfrac{1}{3}\cdot x-\dfrac{5}{3} \\ \\ g(x) = f^{-1}(x) \Rightarrow mx+(m-2) = \dfrac{}{}\dfrac{1}{3}\cdot x -\dfrac{5}{3}
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[tex]\left\{ \begin{array}{ll} m = \dfrac{1}{3} \\ m-2 = -\dfrac{5}{3} \end{array} \right \Rightarrow \left\{ \begin{array}{ll} m = \dfrac{1}{3} \\ m= -\dfrac{5}{3}+2 \end{array} \right \Rightarrow \left\{ \begin{array}{ll} m = \dfrac{1}{3} \\ m = -\dfrac{5}{3}+\dfrac{6}{3}\end{array} \right \Rightarrow \\ \\ \Rightarrow \left\{ \begin{array}{ll} m = \dfrac{1}{3} \\\\ m = \dfrac{1}{3} \end{array} \right| \Rightarrow \boxed{\boxed{m = \dfrac{1}{3}}}[/tex]
