Composite Functions

How do you compose two functions?

Given two functions f[g(x)], step1: substitute the inner function g(x), for x step2: insert into outer function f(x) step3: perform operations step4: combine like terms

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the first step?

Step 1: Substitute g(x) for x f[g(x)] = f[x - 8]

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the second step?

Step 2: Insert into f(x) f[g(x)] = 3(x - 8)² + 6

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the third step?

Step 3: Factor = 3(x² - 16x + 64) + 6

If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the fourth step?

Step 4: combine like terms = 3x² - 48x + 198.

How do you find the domain of a composite function?

Step1: find the domain (restrictions) of the inner function Step2: combine the functions step3: find the domain (restrictions) of the composite function step4: compose domains.

Composition of functions is not commutative. True or false?

True! f[g(x)] is generally not equal to g[f(x)]. Consider f(x) = 2x, and g(x) = x - 3 f[g(x)] = 2(x - 3) = 2x - 6 g[f(x)] = (2x) - 3 = 2x - 3 f[g(x)] is not equal to g[f(x)].

[f o g](x) = f[g(x)] true or false?

True! these are two ways of writing the same thing.

decompose h(x) = (x-4)²

if we choose to remove the component (x-4) as the inner function g(x), then we replace x for every g(x) in f. h(x) = (x-4)² becomes = f[g(x)], or = f(x-4) so f(x) = x² , and g(x) = x - 4 the answer depends on which component you choose to remove.

find the domains of g[f(x)], if f(x) = 4x − 6, and g(x) = √x .

step1: in f(x) = 4x − 6, x is all real numbers step2: in g(f(x)) = √f(x) = √(4x − 6) step3: in g(f(x)) = √(4x − 6) √(4x − 6) ≥ 0, so x ≥ (3/2) step4: in g(f(x)), Domain {x | x ≥ (3/2) }

find the domains of f[g(x)], if f(x)= 3/(x-2) and g(x)=√(1-x)

step1: in g(x) = √(1-x), x ≤ 1 step2: in f[g(x)] = 3/(√(1-x) - 2) step3: in 3/(√(1-x) - 2), √(1-x) ≠ 2 , so (1-x) ≠ 4, finally x ≠ -3 step4: f[g(x)], Domain {x | x ≤ 1, and x ≠ -3 }

function decomposition

when you break into simpler functions from a more complicated function; finding the components of a function.