How do you find the compositions given #g(x) = 3x + 2# and #h(x) = 9x^2 + 12x + 9#?

Question

Liam Brennan · Accepted Answer

#g(h(x))=27x^2+36x+29#
#h(g(x))=81x^2+144x+69# Function compositions are basically plugging one function into another function. To find #g(h(x))#, take #h(x)#, which is #9x^2+12+9#, and plug it into the #x# in #g(x)#. #g(color(blue)(x))=3color(blue)x+2# #h(x)=9x^2+12x+9# #g(color(blue)(h(x)))=3color(blue)((9x^2+12x+9))+2# #g(h(x))=27x^2+36x+27+2# #g(h(x))=27x^2+36x+29# To find #h(g(x))#, do the process with the roles switched: #g(x)# is plugged into #h(x)#. #h(g(x))=9(3x+2)^2+12(3x+2)+9# #h(g(x))=9(9x^2+12x+4)+36x+24+9# #h(g(x))=81x^2+108x+36+36x+33# #h(g(x))=81x^2+144x+69#

Elias Ackerman · Answer

undefined To find the compositions, you would substitute the expression for one function into the other function. For $ g(x) = 3x + 2 $ and $ h(x) = 9x^2 + 12x + 9 $, the compositions would be $ g(h(x)) $ and $ h(g(x)) $.

1. To find $ g(h(x)) $, substitute $ h(x) $ into $ g(x) $, so you would replace $ x $ in $ g(x) $ with $ h(x) $.
   $$ g(h(x)) = 3(9x^2 + 12x + 9) + 2 $$
   Simplify this expression to find $ g(h(x)) $.

2. To find $ h(g(x)) $, substitute $ g(x) $ into $ h(x) $, so you would replace $ x $ in $ h(x) $ with $ g(x) $.
   $$ h(g(x)) = 9(3x + 2)^2 + 12(3x + 2) + 9 $$
   Simplify this expression to find $ h(g(x)) $.