To compose a function is to input one function into the other to form a different function. Here's a few examples.
Example 1: If #f(x) = 2x + 5# and #g(x) = 4x - 1#, determine #f(g(x))#
This would mean inputting #g(x)# for #x# inside #f(x)#.
#f(g(x)) = 2(4x- 1) + 5 = 8x- 2 + 5 = 8x + 3#
Example 2: If #f(x) = 3x^2 + 12 + 12x# and #g(x) =sqrt(3x)#, determine #g(f(x))# and state the domain
Put #f(x)# into #g(x)#.
#g(f(x)) = sqrt(3(3x^2 + 12x + 12))#
#g(f(x)) = sqrt(9x^2 + 36x + 36)#
#g(f(x)) = sqrt((3x + 6)^2)#
#g(f(x)) = |3x + 6|#
The domain of #f(x)# is #x in RR#. The domain of #g(x)# is #x > 0#. Hence, the domain of #g(f(x))# is #x > 0#.
Example 3: if #h(x) = log_2 (3x^2 + 5)# and #m(x) = sqrt(x + 1)#, find the value of #h(m(0))#?
Find the composition, and then evaluate at the given point.
#h(m(x)) = log_2 (3(sqrt(x + 1))^2 + 5)#
#h(m(x)) = log_2 (3(x + 1) + 5)#
#h(m(x)) = log_2 (3x + 3 + 5)#
#h(m(x)) = log_2 (3x + 8)#
#h(m(2)) = log_2 (3(0) + 8)#
#h(m(2)) = log_2 8#
#h(m(2)) = 3#
Practice exercises
For the following exercises: #f(x) = 2x + 7, g(x) = 2^(x - 7) and h(x) = 2x^3 - 4#
a) Determine #f(g(x))#
b) Determine #h(f(x))#
c) Determine #g(h(2))#
Hopefully this helps, and good luck!
