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!
