Transcript 1.6 ppt

Section 1.6
The Algebra of Functions
Algebra of Functions Part 1
• The operations of addition, subtraction,
multiplication, and division can be used to
form new functions from given functions.
Example 1 Combining Functions
• Suppose a small company silk-screens T-shirts with slogans and
logos. The company has costs associated with making the T-shirts
and earns revenue when T-shirts are sold. The cost and revenue
functions are given by
2
C  x   2 4 6  1 .9 x
R ( x ) = 15 x - 0.041x
where x represents the number of T-shirts made and sold. Suppose
further that the company can make between 20 and 300 T-shirts
each week.
a. Find the profit function P  x   R  x   C  x  .
b. Sketch a graph of all three functions on the same coordinate grid.
Combining Functions by Using
Addition, Subtraction, Multiplication
and Division
y   f  g  x 
y
y  f x
10
8
y  g
6
4
2
x
0
2
4
6
8
10
x
Combined Functions
Suppose f and g are functions, then the following
are also functions:
• The sum function, f + g
• The difference function, f - g
• The product function, fg
• The quotient function, f/g
The domains of each of the new functions are the
input values common to both original functions.
The quotient function has the additional restriction
that its domain does not contain the input values
that make the denominator zero.
Example 2 Combining Functions
Numerically
Suppose f and g are functions defined by the
following table of values.
x
-1
0
1
2
3
4
f(x)
6
5
4
3
2
Und.
g(x)
Und.
-1
0
3
8
15
Find the domains of f and g. Identify the domain of each of the
following functions and construct a table of values for each:
f + g, f – g, fg, f/g.
Example 3 Combining Functions
Graphically
• Let f and g be defined by the following graphs. Assume
that the domain of each is the set of real numbers.
5
4
3
2
1
-5 -4 -3 -2 -1-10
-2
-3
-4
-5
y
y  f x
5
4
3
2
1
x
1
2 3
4 5
-5 -4 -3 -2 -1-10
-2
-3
-4
-5
y
y  g x
x
1 2 3 4 5
Find the following values, if possible. If a value is undefined, state so and
explain why. a. (f + g)(0) b. (g -f)(0)
Example 4 Combining Functions
Symbolically
f x  x  1
Let
and g  x   x  1 ,
which are both defined for all real numbers.
Find the symbolic representation of following
functions and state the domain of each.
a. f + g b. f – g c. fg d. f/g e. g/f
2
Applications of Combining
Functions
• Suppose x is the number of items made. The following functions are
found by using basic functions
• Total Costs: The total amount of money associated with making x
items.
C  x   Fixed costs + variable costs
• Revenue: The total amount of money received from the sale of x
items.
R(x) = (number of items sold)(price per item)
• Profit: The amount that revenue exceeds total costs. Negative
profit is called loss
P(x) = R(x) – C(x)
Unit Cost: The cost per unit when x items are made. C(x)/x
• Unit Profit: The profit per unit when x items are sold. P(x)/x
Example 5 An Application of
Combining Functions
• Suppose a manufacturer has fixed costs of
$5000, variable costs of $4 for each unit
made, and sells the product for $20. Write
a function that represents the following.
• a. Total cost
• b. Unit cost
• c. Revenue
• d. Profit
• e. Unit profit
Composition of Functions
• Another way of combining functions is to use the
output value of one function as the input value of
a second function.
• For example, suppose you are taking a trip and
you will be visiting two countries, Japan and then
Mexico. You might use a function to convert your
dollars to yen in Japan and then another
function to convert your yen to pesos in Mexico.
Our composite function would be a single
function that would convert the input in dollars
directly to pesos.
Dollars
Yen
g
f of g
Pesos
f
f
g
Composition of Functions
• The new operation is called composition
and it is written as f g , which is read “f
composed with g.” The result of
composition is a new function that yields
the result of finding the first output, g(x),
and then using that output as the input of f
to find  f g  ( x ) , so another way to write
the composition of f with g is f[g(x)].
Example 6 Composition of
Functions Numerically
x
0
1
2
3
f(x)
4
3
2
1
g(x)
1
2
0
3
• Use the table of values to find the indicated
function values. If a value is undefined, explain
why.
• a. f(g(2)) b. f(g(1)) c. f(g(3)) d. g(f(1))
• e. g(f(2)) f. g(f(2)) g. f(f(1)) h. g(g(-4))
Composition of Functions
Graphically
• Composition of two functions can also be
found by using graphs of the two functions
to find values.
Example 7 Composition of
Functions Graphically
• Suppose that f and g
are defined by the
graphs below.
• a.  f g  (5 )
• b.  g f  (0 )
• c.  g f  (3 )
• d.  g f  ( 2 )
• e.  f f  ( 4 )
f
g  (0 )
Example 8 Applying
Composition of Functions
Suppose a circular puddle is slowly evaporating
and the radius of the puddle (in inches) is
r t  
20
3t  1
t minutes after it stops raining. The area of the
puddle is given by A  r    r 2 .
• Find a function that gives the area of the puddle
t minutes after it stops raining.
• Find the area of the puddle 10 minutes after it
stops raining.
Composition of Inverse
Functions
• When both orderings of a relation are
functions and inverse functions are
composed, the result is the original input.
That is, suppose y  f  x  and x  f  1  y 
f
f
1
y   y
and

f
1
f
x 
x.
Example 9 Composition of
Inverse Functions
• Suppose y = f(x) = 3x + 4 and x = g(y) = y/3 + 4/3.
Find the following:
• a.  f g  ( x )
• b. ( g f )( x )
• c. Determine whether the functions are inverses of
each other