1. 3.3 Perform Function Operations & Composition
What is the difference between a power function and a polynomial equation?
What operations can be performed on functions?
What is a composition of two functions?
How is a composition of functions evaluated?

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. Power Functions

4. Domain of (a) all real numbers Domain of (b) all real numbers
Ex: 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

5. Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.
f(x) + g(x)
SOLUTION
f (x) + g(x)
= [4 + (–9)]x1/2
= –5x1/2 b.
f(x) – g(x)
SOLUTION
f (x) – g(x)
= 4x1/2 – (–9x1/2)
= [4 – (–9)]x1/2
= 13x1/2 c.
the domains of f + g and f – g
The functions f and g each have the same domain: all nonnegative real numbers. So, the domains of f + g and f – g also consist of all nonnegative real numbers.

6. Types Domains
All real numbers – if you can use positive numbers, negative numbers, or zero in the beginning functions and the result of combining functions.
All non-negative numbers -- if you can use positive numbers and zero in the beginning functions and the result of combining functions.
All positive numbers -- if you can use only positive numbers in the beginning function and the result of combining functions

7. Ex: Let f(x)=4x1/3 & g(x)=x1/2
Ex: 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 (Non-neg #’s).
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 (Positive #’s).

8. Let f (x) = 6x and g(x) = x3/4. Find the following.
SOLUTION
f (x)
g(x) = 6x
x3/4 =
6x(1 – 3/4) =
6x1/4
SOLUTION
the domain of
The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers.

9. Try it… Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.
f (x) + g(x)
SOLUTION
f (x) + g(x)
= –2x2/3 + 7x2/3
= (–2 + 7)x2/3
= 5x2/3
f (x) – g(x)
SOLUTION
f (x) – g(x)
= –2x2/3 – 7x2/3
= [–2 + ( –7)]x2/3
= –9x2/3
The domains of f and g have the same domain: all non-negative real numbers. So , the domain of f + g and f – g also consist of all non-negative real numbers.

10. Composition of Functions

11. Ex: Let f(x)=2x-1 & g(x)=x2-1
Ex: 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.

12. Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
g(f(5))
SOLUTION
To evaluate g(f(5)), you first must find f(5).
f(5)
= 3(5) – 8
= 7
Then g( f(3))
= g(7)
= 2(7)2
= 2(49)
= 98.
So, the value of g(f(5)) is 98.
ANSWER

13. Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
f(g(5))
SOLUTION
To evaluate f(g(5)), you first must find g(5).
g (5)
= 2(5)2
= 2(25)
= 50
Then f( g(5))
= f(50)
= 3(50) – 8
= 150 – 8
= 142.
So, the value of f( g(5)) is 142.
ANSWER

14. Let f(x) = 2x–1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)), and f(f(x))
Let f(x) = 2x–1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)), and f(f(x)). Then state the domain of each composition.
SOLUTION = 2
2x + 7
f(g(x)) =
f(2x + 7)
= 2(2x + 7)–1
g(f(x)) =
f(2x–1) 4 x =
+ 7
= 2(2x–1) + 7
= 4x–1 + 7
f(f(x)) =
f(2x–1)
= 2(2x–1)–1
= x
The domain of f(g(x )) consists of all real numbers except x = –3.5. The domain of g(f(x)) consists of all real numbers except x = 0.

15. What is the difference between a power function and a polynomial equation?
The power tells you what kind of equation—linear, quadratic, cubic…
What operations can be performed on functions?
Add, subtract, multiply, divide.
What is a composition of two functions?
A equation (function) is substituted in for the x in another equation (function).
How is a composition of functions evaluated?
Write the outside function and substitute the other function for x.

16. Page 184, 3-27 every 3rd problem, 29-35 odd
Assignment
Page 184, 3-27 every 3rd problem, odd