BY

DR. JULIA ARNOLD

The Algebra of Functions

The Algebra of Functions

What does it mean to add two functions?If f(x) = 2x + 3 and g(x) = -4x - 2What would (f+g)(x) be?

(f+g)(x) = f(x) + g(x) It means to add the two functions(f+g)(x) = -2x + 1

Likewise (f - g)(x) = f(x) - g(x) or (f - g)(x) = 6x +5

Multiplication of two functions is expressed like this:fg(x) = f(x)g(x)In our example, fg(x) = -8x2 -16x -6

The Algebra of Functions

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

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

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

Division also follows a logical path:

0xgxgxf

xgf

)(,

In our example: f(x) = 2x + 3 and g(x) = -4x - 2

02x4

2x43x2

xgxf

xgf

,

Application:In business you would have fixed costs, such as rent, and variable costs from producing your commodity.We will call the cost C(x)

Revenue is the money you make in your business. We will call revenue R(x).

Profit is what you hope to make from your business and is denoted as P(x) = R(x) - C(x).

Suppose your company manufactures water filters and has fixed costs of $10,000 per month. The cost of producing the water filters is represented by -.0001x2 + 10x where How do you represent the cost function?C(x) = -.0001x2 + 10x +10000

00040x0 ,00040x0 ,

Suppose the total revenue you make from the sale of x water filters is given by R(x) = -.0005x2 + 20x 00040x0 ,

What would the profit function be?

How much would you make if you sold 10,000 water filters?

Suppose your company manufactures water filters and has fixed costs of $10,000 per month. The cost of producing the water filters is represented by -.0001x2 + 10x where How do you represent the cost function?C(x) = -.0001x2 + 10x +10000

00040x0 ,00040x0 ,

Suppose the total revenue you make from the sale of x water filters is given by R(x) = -.0005x2 + 20x 00040x0 ,

What would the profit function be?P(x) = R(x) - C(x) = -.0005x2 + 20x - (-.0001x2 + 10x +10000)P(x)= (-.0005 +.0001) x2 +20x -10x - 10000 P(x) = -.0004x2 +10x -10000

How much would you make if you sold 10,000 water filters?P(10000)= -.0004(10000)2 +10(10000) - 10000 = 50,000 per month

Composition of Functions (one more operation)The easiest way to describe composition is to say it is like substitution. In fact

))(()( xgfxgf

Read f of g of x which means substitute g(x) for x in the f(x) expression.

Suppose f(x)= 2x + 3 and g(x) = 8 - x

f(g(x) )= 2 g(x) + 3

f(8 - x)= 2 (8 - x) + 3f(g(x)) = 16 -2x + 3 or 19 - 2x

An interesting fact is that

xfgxgf most of the time.

Let’s see if this is thecase for theprevious example.

f(x) = 2x + 3, and g(x) = 8 - x

))(( xfgxfg Thus we will substitute f into g.

g(x) = 8 - x

g(f(x) ) = 8 - f(x)

Nowsubstitutewhat f(x) is:

g(2x + 3) = 8 - (2x + 3)= 8 - 2x - 3= 5 - 2x

f(g(x)) = 19 - 2x while g(f(x)= 5 - 2x

32)( 2 xxxf xxg )(

))(()( xgfxgf

Write the f function 32)( 2 xxxfSubstitute g(x) for x

3)(2)())(( 2 xgxgxgf

Replace g(x) with x 32)(2

xxxf

Simplify32))(( xxxgf

Step 1

Step 2

Step 3

Step 4

32)( 2 xxxf xxg )(

Find: xfg

When ready click your mouse.

The answer is:

A) 322 xx

B) 32 xxMove your mouse overthe correct answer.

xxxf 2)(xx

xg

2

1)(

Find: xgf

When ready click your mouse.

The answer is:

B) xxxx

2

2

2

11

A)xxxx

224

11

Move your mouse overthe correct answer.

1)( xxf 21

)( x

xg

Find: xfg

When ready click your mouse.

The answer is:

B) 2

1

1

x

A) 121

121

x

x

xMove your mouse overthe correct answer.

We can also evaluate the composition of functions at a number.Let:

1)( xxf

xxxg

2

1)(

and

Find 3gf

1)( xxfxx

xg

2

1)(

Find 3gf = f(g(3))

This says to insert the value for g(3)into f, so…Step 1 is to find g(3)

12

1

39

1

33

1)3(

2

g

1)( xxfxx

xg

2

1)(

Find 3gf = f(g(3))

Now substitute the answer into f(x) for x.

12

1)3( g

6

631

6

31

36

31

3

3

12

11

12

11

12

1

12

1

f

1 2 3 4 51: Take the square root of top and bottom.

2: Find a number that rationalizes the denominator

3: multiply top and bottom 4: Take the square root of 36

5: add the 1 as 6/6