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