Objective: Perform Operations on Functions

Objective: Perform Operations on Functions

Basic Operations on Functions

You can add, subtract, multiply, and divide functions in much the same way as you do with real numbers.

Given two function f and g, for all x in the domain of both f and g the basic operations are defined as follows:

( f + g)(x) = f (x) + g(x) ( f - g)(x) = f (x) - g(x) ( f g)(x) = f (x) g(x) ( f / g)(x) = f (x) , for g(x) 0

g(x)

Consider f (x) = x2 + 2x - 3 and g(x) = x -1

( f + g)(x) = f (x) + g(x) = x2 + 2x -3+ x -1 = x2 + 3x - 4

You can check with your calculator by graphing the sumY1+Y2. You can find Y1 and Y2. Under VARS, FUNCTIONS.

( f - g)(x) = f (x) - g(x)

( ) = x2 + 2x - 3 - (x -1)

= x2 + x - 2

You can check with your calculator by graphing the difference, Y1-Y2.

Objective: Perform Operations on Functions

( f g)(x) = f (x) g(x)

( ) = x2 + 2x - 3 (x -1)

= x2 (x -1) + 2x(x -1) - 3(x -1) = x3 - x2 + 2x2 - 2x - 3x + 3 = x3 + x2 - 5x + 3

You can check with your calculator by graphing the product, Y1*Y2.

( f / g)(x) = f (x) g(x)

= x2 + 2x - 3 x -1

= (x + 3)(x -1) x -1

= x+3

Note that even though the denominator divides the numerator, ( f / g)(x) is not defined for x = 1, as shown by the "hole" in the graph.

Objective: Perform Operations on Functions

Composition of Functions

The composition of two functions f and g, denoted f o g is defined: ( f o g)(x) = f (g(x))

for all x in the domain of g such that g(x) is in the domain of f.

Consider f (x) = -16x2 + 80x and g(x) = x + 3

( f o g)(x) = f (g(x)) = f (x + 3) = -16(x + 3)2 + 80(x + 3) = -16x2 - 96x -144 + 80x + 240 = -16x2 -16x + 96

You can check with your calculator by graphing the composition Y1(Y2).

Note the result is a horizontal shift of f (x) to the left 3 units. The maximum value remains 100.

Note that g o f does not yield the same function. (g o f )(x) = g( f (x)) = g(-16x2 + 80x) = -16x2 + 80x + 3

You can check with your calculator by graphing the composition Y2(Y1).

Note the result is a vertical shift of f (x) up 3 units. The maximum value is 103.

Objective: Perform Operations on Functions

Decomposition of Functions

You can often rewrite a function as the composition of two functions more than one way.

Consider h(x) = 2x + 1 . Find two functions f and g, such that, h(x) = ( f o g)(x) . Solution 1:

Let f (x) = x and g(x) = 2x +1. Then h(x) = ( f o g)(x) = f (g(x)) = f (2x + 1) = 2x + 1 .

Solution 2:

Let f (x) = 2x and g(x) = x + 1 . 2

Then

h(x)

=

(

f

o

g )( x)

=

f

( g ( x))

=

f

x

+

1 2

=

2 x + 1 = 2

2x +1.

Solution 3:

Let f (x) = x and g(x) = 2x + 1

( ) Then h(x) = ( f o g)(x) = f (g(x)) = f 2x + 1 = 2x + 1 .

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