1. 6.3 Perform Function Operations & Composition
Algebra II

2. Operations on Functions: for any two functions f(x) & g(x)
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.)

3. Domain of (a) all real numbers Domain of (b) all real numbers
Ex. 1: Let f(x)=3x1/3 & g(x)=2x1/3. Find (a) the sum, (b) the difference, and (c) the domain for each.
3x1/3 + 2x1/3
= 5x1/3
3x1/3 – 2x1/3
= x1/3
Domain of (a) all real numbers
Domain of (b) all real numbers

4. Ex. 2: Let f(x)=4x1/3 & g(x)=x1/2
Ex. 2: Let f(x)=4x1/3 & g(x)=x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each.
4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6
= 4x1/3-1/2
= 4x-1/6 =
(c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number.
Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero.

5. 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.

6. Evaluate Compositions of functions Ex. 3
Let f(x) =3x – 4 and g(x) =
What is the value of f(g(-3))?

7. Ex. 4: Let f(x)=2x-1 & g(x)=x2-1
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 =
(c) 2(2x-1)-1
= 2(2-1x) =
(b) (2x-1)2-1
= 22x-2-1 =
(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.

8. Ex. 5 Find Composition of Functions
Perform the indicated operation & state the domain.
Let g(x) =2x -7 &
5a.) g(h(x)) 5b.) g(g(x))
4a) #31) 2x-13/3 4b) # 35) 4x-21

9. Assignment

10. 6-3B Compositions