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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