For each pair of functions, write down the composite functions fg(x) and gf(x) a f(x)=x+6 g(x)=x-3 b f(x)=2x^2-3x+1 g(x)=5x C f(x)=3x^2-4x+2 g(x)=3x-2 d f(x)=(4x)/(3) g(x)=x^2-9

a) For the functions $f(x)=x+6$ and $g(x)=x-3$, the composite functions are:<br />$fg(x) = f(g(x)) = f(x-3) = (x-3) + 6 = x + 3$<br />$gf(x) = g(f(x)) = g(x+6) = (x+6) - 3 = x + 3$<br /><br />b) For the functions $f(x)=2x^{2}-3x+1$ and $g(x)=5x$, the composite functions are:<br />$fg(x) = f(g(x)) = f(5x) = 2(5x)^{2} - 3(5x) + 1 = 50x^{2} - 15x + 1$<br />$gf(x) = g(f(x)) = g(2x^{2}-3x+1) = 5(2x^{2}-3x+1) = 10x^{2} - 15x + 5$<br /><br />c) For the functions $f(x)=3x^{2}-4x+2$ and $g(x)=3x-2$, the composite functions are:<br />$fg(x) = f(g(x)) = f(3x-2) = 3(3x-2)^{2} - 4(3x-2) + 2 = 27x^{2} - 36x + 14$<br />$gf(x) = g(f(x)) = g(3x^{2}-4x+2) = 3(3x^{2}-4x+2) - 2 = 9x^{2} - 12x + 4$<br /><br />d) For the functions $f(x)=\frac{4x}{3}$ and $g(x)=x^{2}-9$, the composite functions are:<br />$fg(x) = f(g(x)) = f(x^{2}-9) = \frac{4(x^{2}-9)}{3} = \frac{4x^{2}}{3} - 12$<br />$gf(x) = g(f(x)) = g(\frac{4x}{3}) = (\frac{4x}{3})^{2} - 9 = \frac{16x^{2}}{9} - 9$