1. The Algebra of Functions
Section 1.6
The Algebra of Functions

2. Algebra of Functions Part 1
The operations of addition, subtraction, multiplication, and division can be used to form new functions from given functions.

3. 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
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
b. Sketch a graph of all three functions on the same coordinate grid.

4. Combining Functions by Using Addition, Subtraction, Multiplication and Division

5. 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.

6. Example 2 Combining Functions Numerically
Suppose f and g are functions defined by the following table of values. x -1 1 2 3 4
f(x) 6 5
Und.
g(x) 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.

7. 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.
Find the following values, if possible. If a value is undefined, state so and explain why. a. (f + g)(0) b. (g -f)(0)

8. Example 4 Combining Functions Symbolically
Let and ,
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

9. 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.
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

10. 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

11. 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.

12. Dollars Yen Pesos
g f
f of g

13. Composition of Functions
The new operation is called composition and it is written as , 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 , so another way to write the composition of f with g is f[g(x)].

14. Example 6 Composition of Functions Numerically1 2 3
f(x) 4
g(x)
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))

15. Composition of Functions Graphically
Composition of two functions can also be found by using graphs of the two functions to find values.

16. Example 7 Composition of Functions Graphically
Suppose that f and g are defined by the graphs below. a. b. c. d. e.

17. Example 8 Applying Composition of Functions
Suppose a circular puddle is slowly evaporating and the radius of the puddle (in inches) is
t minutes after it stops raining. The area of the puddle is given by
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.

18. 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 and
and

19. Example 9 Composition of Inverse Functions
Suppose y = f(x) = 3x + 4 and x = g(y) = y/3 + 4/3. Find the following: a. b.
c. Determine whether the functions are inverses of each other