The Algebra of Functions
BY
DR. JULIA ARNOLD
The Algebra of Functions
What does it mean to add two functions?
If f(x) = 2x + 3 and g(x) = -4x - 2
What 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:
f
g
x  
f x 
g x 
, g (x )  0
In our example: f(x) = 2x + 3 and g(x) = -4x - 2
f
g
x  
f x 
g x 

2x  3
 4x  2
, 4 x  2  0
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
0  x  40 , 000 How do you represent the cost function?
0  x  40 , 000
C(x) = -.0001x2 + 10x +10000
Suppose the total revenue you make from the sale of x water
filters is given by R(x) = -.0005x2 + 20x 0  x  40 , 000
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
0  x  40 , 000
How do you represent the cost function?
0  x  40 , 000
C(x) = -.0001x2 + 10x +10000
Suppose the total revenue you make from the sale of x water
filters is given by R(x) = -.0005x2 + 20x 0  x  40 , 000
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
f
 g ( x )  f ( g ( x ))
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) + 3
f(g(x)) = 16 -2x + 3 or 19 - 2x
An interesting fact is that
f
 g  x    g  f
Let’s see if
this is the
case for
the
previous
example.
 x 
most of the time.
f(x) = 2x + 3, and
g(x) = 8 - x
g  f x   g ( f ( x ))
Thus we will substitute f into g.
g(x) = 8 - x
g(f(x) ) = 8 - f(x)
Now
substitute
what 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
f ( x)  x  2 x  3
g ( x) 
2
f
 g ( x )  f ( g ( x ))
Step 1
Write the f function
Step 2
Substitute g(x) for x
Step 3
Replace g(x) with
f ( x)  x  2 x  3
2
f ( g ( x ))  g ( x )  2 g ( x )  3
2
x
f ( x) 
Step 4
Simplify
x
 x 2
f ( g ( x ))  x  2 x  3
2
x 3
f ( x)  x  2 x  3
2
Find:
g ( x) 
x
 g  f  x 
When ready click your
mouse.
Move your mouse over
the correct answer.
The answer is:
A)
x  2x  3
2
B) x  2 x  3
g ( x) 
f ( x)  x  x
2
Find:
f
 g  x 
1
x x
2
The answer is:
When ready click your
mouse.
Move your mouse over
the correct answer.
A)
1
x x
4
2

2
1
x x
2
1
 1 
B)  2
  2
x x
x x
f (x) 
Find:
g ( x) 
x 1
1
2
x
 g  f  x 
The answer is:
When ready click your
mouse.
1
A)
Move your mouse over
the correct answer.
 2 1 
x
B)
1 2x
x
1
x 1
2
1
We can also evaluate the
composition of functions at a
number.
Let:
f (x) 
x 1
g ( x) 
1
and
Find
f
x x
2
 g 3 
f (x) 
Find
f
x 1
g ( x) 
 g 3  = f(g(3))
1
x x
2
This says to insert the value for g(3)
into f, so…
Step 1 is to find g(3)
g (3) 
1
3 3
2

1
93

1
12
f (x) 
Find
f
1
g ( x) 
x 1
x x
 g 3  = f(g(3))
2
g (3) 
1
12
Now substitute the answer into f(x)
for x.
1
1
1
1 
f 
1 
1 
12
12
12
 12 
1
3
3
2
1 
3
36
1 
3
3
6
1 
4
36
6
5
1: Take the square root of top and bottom. 5: add the 1 as 6/6
2: Find a number that rationalizes the denominator
3: multiply top and bottom
4: Take the square root of 36