How do you find the compositions given #f(x)=3x²# and #g(x)=4-5x#?

The example I'll show you is #f(g(x))#. This means plugging in function g for x into f.

#f(g(x)) = 3(4 - 5x)^2#

#f(g(x)) = 3(16 - 40x + 25x^2)#

#f(g(x)) = 48 - 120x + 75x^2#

#f(g(x)) = 75x^2 - 120x + 48#

Note that function compositions will often be noted as #(f @ g)(x)#, which means the same thing as #f(g(x))# (it must be evaluated from the inside outside; if you have #f(g(h(x)))# for example, you must plug h into g and then plug the answer of that into f.)

When the x in parentheses is replaced by a number, you must plug this in for x.

Ex: if #f(x) = 3x + 4# and #g(x) = 3^x#, find the value of #f(g(2))#.

#f(g(2) = 3(3^2) + 4#

#f(g(2)) = 27 + 4#

#f(g(2)) = 31#

Practice exercises:

1. Find the following compositions if #f(x) = 2x + 5, g(x) = 2x^2 - 4x + 7 and h(x) = sqrt(5x - 1)#

a) #h(f(x))#

b). #h(g(f(x)))#

c). #g(f(h(13)))#

Good luck!

(1) find f(g(x)) = f(4 - 5x)

To find the value , substitute x = 4 - 5x in for x in f(x)

hence : f(4-5x) = 3(4-5x)^2 = # 3(16-40x+25x^2) #

# = 48-120x +75x^2 = 75x^2 -120x + 48 #

(2) find g(f(x)) # = g(3x^2) #

To find the value , substitute x = #3x^2" in for x in g(x) "#

hence : #g(3x^2) = 4 - 5(3x^2) = 4 - 15x^2#