Operations With & Composite Functions

to compose functions

evaluate outer function using the inner function

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.

to multiply polynomial functions

apply the distributive property

to add radical functions

combine coefficients in front of radical with common index and radicand

to add polynomial functions

combine like terms

to subtract polynomial functions

distribute negative one to subtrahend and combine like terms

to add rational functions

get a common denominator and equivalent numerators, then combine like terms in numerators

to subtract rational functions

get common denominator and equivalent numerators, distribute negative one to the subtrahend, then combine like terms in numerators

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.

to multiply rational functions

multiply numerators and multiply denominators

to divide rational functions

multiply the dividend by the reciprocal of the divisor

criteria for multiplying radical functions

must have same index

criteria for adding radical functions

must have save index and radicand

to multiply radical functions

product of coefficients is placed in front of radical with common index, multiply radicands

to divide polynomial functions

set dividend over divisor, factor both with respect to integers, cancel common factors, simplify

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.