6.3 Perform Function
Operations & Composition
Algebra II
Operations on Functions: for any
two functions f(x) & g(x)
1.
2.
3.
4.
5.
**
Addition
h(x) = f(x) + g(x)
Subtraction h(x) = f(x) – g(x)
Multiplication h(x) = f(x)*g(x) OR f(x)g(x)
Division
h(x) = f(x)/g(x) OR f(x) ÷ g(x)
Composition h(x) = f(g(x)) OR g(f(x))
Domain – all real x-values that “make sense”
(i.e. can’t have a zero in the denominator,
can’t take the even nth root of a negative
number, etc.)
Ex. 1: Let f(x)=3x1/3 & g(x)=2x1/3. Find
(a) the sum, (b) the difference, and (c)
the domain for each.
(a) 3x1/3 + 2x1/3
= 5x1/3
(b) 3x1/3 – 2x1/3
= x1/3
(c) Domain of (a) all real numbers
Domain of (b) all real numbers
Ex. 2: Let f(x)=4x1/3 & g(x)=x1/2. Find
(a) the product, (b) the quotient, and (c)
the domain for each.
5
(a) 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6
(b)
4x
x
1
3
 x
6
(c) Domain of (a) all reals ≥ 0,
because you can’t take the 6th root of
a negative number.
1
2
= 4x1/3-1/2
= 4x-1/6
= 4
x
4
Domain of (b) all reals > 0,
because you can’t take the 6th root of
a negative number and you can’t have
4 a denominator of zero.
1
6
6
x
Composition
• f(g(x)) means you take the function g and
plug it in for the x-values in the function f,
then simplify.
• g(f(x)) means you take the function f and
plug it in for the x-values in the function g,
then simplify.
Evaluate Compositions of
functions Ex. 3
• Let f(x) =3x – 4 and g(x) = x  1
• What is the value of f(g(-3))?
2
Ex. 4: Let f(x)=2x-1 & g(x)=x2-1. Find (a)
f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the
domain of each.
(a) 2(x2-1)-1 =
(b) (2x-1)2-1
= 22x-2-1
4
= 2 1
x
2
x2 1
(c) 2(2x-1)-1
= 2(2-1x)
= 2x  x
2
(d) Domain of (a) all reals except
x=±1.
Domain of (b) all reals except x=0.
Domain of (c) all reals except x=0,
because 2x-1 can’t have x=0.
Ex. 5 Find Composition of
Functions
• Perform the indicated operation & state
the domain.
x4
• Let g(x) =2x -7 & h( x ) 
3
• 5a.) g(h(x))
5b.) g(g(x))
Assignment
6-3B Compositions
•