[tex] \begin{align} (g\circ f) (x) &= 4x^2+12x \\ g(f(x)) &= 4x^2+12x \\ 2(f(x)) +1 &= 4x^2+12x \\ 2(f(x)) &= 4x^2+12x-1 \\ f(x) &= 2x^2+6x- \dfrac{1}{2} \end{align} [/tex]
[tex] \begin{align} (f\circ g)(x) &= f(g(x)) \\ &= f(2x+1) \\ &= 2(2x+1)^2+6(2x+1)- \dfrac12 \\ &= 2(4x^2+4x+1)+12x+6- \dfrac12 \\ &= 8x^2+8x+2+12x+6- \dfrac12 \\ &= 8x^2+20x+ \dfrac{15}{2}\end{align} [/tex]
[tex] \begin{align} (f\circ g)(2) &= 8(2)^2+20(2)+ \dfrac{15}{2} \\ &= 32+40+ \dfrac{15}{2} \\ &= \dfrac{159}{2} \end{align} [/tex]
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[tex] \begin{align} (g\circ f) (x) &= 4x^2+12x \\ g(f(x)) &= 4x^2+12x \\ 2(f(x)) +1 &= 4x^2+12x \\ 2(f(x)) &= 4x^2+12x-1 \\ f(x) &= 2x^2+6x- \dfrac{1}{2} \end{align} [/tex]
[tex] \begin{align} (f\circ g)(x) &= f(g(x)) \\ &= f(2x+1) \\ &= 2(2x+1)^2+6(2x+1)- \dfrac12 \\ &= 2(4x^2+4x+1)+12x+6- \dfrac12 \\ &= 8x^2+8x+2+12x+6- \dfrac12 \\ &= 8x^2+20x+ \dfrac{15}{2}\end{align} [/tex]
[tex] \begin{align} (f\circ g)(2) &= 8(2)^2+20(2)+ \dfrac{15}{2} \\ &= 32+40+ \dfrac{15}{2} \\ &= \dfrac{159}{2} \end{align} [/tex]