A)
[tex]\\\\\text{Given that,} ~ g(x) = \dfrac 13 x + 5 \\\\\text{Write g(x) as}~ y = \dfrac 13 x +5 \\\\\text{Replace x with y and then solve for y:}\\\\~~~~~~~x = \dfrac 13y +5 \\\\\implies 3x = y + 15~~~~~~~~~~~~~~~~~~~;[\text{Multiply both sides by 3]}\\\\\implies y = 3x -15\\\\\implies g^{-1} (x) = 3x -15 \\[/tex]
B)
[tex]\left(g \circ} g^{-1} \right)(x)\\\\=g\left(g^{-1}(x) \right)\\\\=g(3x-15)\\\\=\dfrac 13\cdot(3x-15)+5\\\\=x -5 +5\\\\=x[/tex]
