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3.3 Perform Function
Operations & Composition
p. 180
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?
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.)
Power Functions
Ex: 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
Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.
a.
f(x) + g(x)
SOLUTION
f (x) + g(x)
= [4 + (–9)]x1/2 = –5x1/2
b. f(x) – g(x)
SOLUTION
= [4 – (–9)]x1/2
f (x) – g(x) = 4x1/2 – (–9x1/2)
c. the domains of f + g and f – g
= 13x1/2
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.
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
Ex: 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 (Non-neg #’s).
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 (Positive #’s).
1
6
6
x
Let f (x) = 6x and g(x) = x3/4. Find the following.
f (x)
g(x)
SOLUTION
f (x)
g(x)
SOLUTION
6x
= x3/4
= 6x(1 – 3/4) = 6x1/4
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.
Try it…
Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.
1. f (x) + g(x)
SOLUTION
f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3
2. 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.
Composition of Functions
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 =
(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.
Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
8. 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.
ANSWER So, the value of g(f(5)) is 98.
Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
9. 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.
ANSWER So, the value of f( g(5)) is 142.
12. 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
f(g(x)) =f(2x + 7) = 2(2x + 7)–1 =
2
2x + 7
g(f(x)) =f(2x–1) = 2(2x–1) + 7 = 4x–1 + 7 = 4 + 7
x
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.
• 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.
Assignment
Page 184, 3-27 every
3rd problem, 29-35 odd