3.3 perform function operations & composition p. 180 what is the difference between a power...
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3.3 Perform Function 3.3 Perform Function Operations & CompositionOperations & 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 FunctionsOperations on Functions: for any : for any two functions f(x) & g(x)two functions f(x) & g(x)
1.1. AdditionAddition h(x) = f(x) + g(x)
2.2. SubtractionSubtraction h(x) = f(x) – g(x)
3.3. MultiplicationMultiplication h(x) = f(x)*g(x) OR f(x)g(x)
4.4. DivisionDivision h(x) = f(x)/g(x) OR f(x) ÷ g(x)
5.5. CompositionComposition h(x) = f(g(x)) OR g(f(x))
** DomainDomain – 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)=3xEx: Let f(x)=3x1/31/3 & g(x)=2x & g(x)=2x1/31/3. Find (a) . Find (a) the sum, (b) the difference, and (c) the the sum, (b) the difference, and (c) the
domain for each.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
SOLUTION
Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.
a. f(x) + g(x)
f (x) + g(x) = [4 + (–9)]x1/2 = –5x1/2
b. f(x) – g(x)
SOLUTION
f (x) – g(x) = [4 – (–9)]x1/2 = 13x1/2= 4x1/2 – (–9x1/2)
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.
c. the domains of f + g and f – g
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)=4xEx: Let f(x)=4x1/31/3 & g(x)=x & g(x)=x1/21/2. Find (a) . Find (a) the product, (b) the quotient, and (c) the the product, (b) the quotient, and (c) the
domain for each.domain for each.
(a) 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6
(b)
= 4x1/3-1/2 = 4x-1/6 =
2
1
3
1
4
x
x
6
1
4
x
(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).
564 x
6
4
x
Let f (x) = 6x and g(x) = x3/4. Find the following.
f (x)g(x)
f (x)g(x) =
6xx3/4 = 6x(1 – 3/4) = 6x1/4
SOLUTION
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.
Try it…Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.
f (x) + g(x)1.
SOLUTION
f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3
f (x) – g(x)2.
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 FunctionsComposition of Functions
Ex: Let f(x)=2xEx: Let f(x)=2x-1-1 & g(x)=x & g(x)=x22-1. Find (a) -1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the
domain of each.domain of each.(a) 2(x2-1)-1 =
1
22 x
(b) (2x-1)2-1
= 22x-2-1
= 142
x
(c) 2(2x-1)-1
= 2(2-1x)
=2
2x x
(d) Domain of (a) all reals except Domain of (a) all reals except x=x=±1.±1.
Domain of (b) all reals except x=0.Domain of (b) all reals except x=0.
Domain of (c) all reals except x=0, Domain of (c) all reals except x=0, because 2xbecause 2x-1-1 can’t have x=0. 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)
Then g( f(3))
= 3(5) – 8 = 7
= g(7) = 2(7)2 = 2(49) = 98.
So, the value of g(f(5)) is 98.ANSWER
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)
Then f( g(5))
= 2(5)2 = 2(25)
= f(50) = 3(50) – 8 = 150 – 8 = 142.
So, the value of f( g(5)) is 142.ANSWER
= 50
SOLUTION
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.
f(g(x))=f(2x + 7) = 2(2x + 7)–1 =2
2x + 7
=f(2x–1) = 2(2x–1) + 7g(f(x)) = 4x–1 + 7 4x = + 7
=f(2x–1) = 2(2x–1)–1f(f(x)) = 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.
AssignmentPage 184, 3-27 every
3rd problem, 29-35 odd