2. Chapter 2: Analysis of Graphs of Functions
2.1 Graphs of Basic Functions and Relations; Symmetry
2.2 Vertical and Horizontal Shifts of Graphs
2.3 Stretching, Shrinking, and Reflecting Graphs
2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications
2.5 Piecewise-Defined Functions
2.6 Operations and Composition

3. 2.6 Operations and Composition
The domain of is the intersection of the domains of f and g, while the domain of f /g is the intersection of the domains of f and g for which
Operations on Functions
Given two functions f and g, then for all values of x for which both and are defined, the functions are defined as follows.
Sum
Difference
Product
Quotient

4. 2.6 Examples Using Operations on Functions
Analytic Solution
(a)
(b)
(c)
(d)

5. 2.6 Graphing Calculator Capabilities of Function Notation
We can support the analytic solution of the previous example with the calculator by using its function notation capability.
Enter f as and g as

6. 2.6 Examples Using Operations on Functions
Solution
(a)
(b)
(c)

7. 2.6 Evaluating Combinations of Functions
If possible, use the given graph of f and g to evaluate
(a)
(b)
(c)
Solution

8. 2.6 The Difference Quotient
The expression is called the difference quotient.
Figure 67 pg 2-153

9. 2.6 Looking Ahead to Calculus
The difference quotient is essential in the definition of the derivative of a function.
the slope of the secant line is an average rate of change
The derivative is used to find the slope of the tangent line to the graph of a function at a point.
the slope of the tangent line is an instantaneous rate of change
The derivative is found by letting h approach zero in the difference quotient.
i.e. the slope of the secant line approaches the slope of the tangent line as h gets close to zero

10. 2.6 Finding the Difference Quotient
Let Find the difference quotient and simplify.
Solution

11. 2.6 Composition of Functions
If f and g are functions, then the composite function, or composition, of g and f is
for all x in the domain of f such that is in the domain of g.

12. 2.6 Application of Composition of Functions
Suppose an oil well off the California coast is leaking.
Leak spreads in circular layer over water
Area of the circle is
At any time t, in minutes, the radius increases
5 feet every minute.
Radius of the circular oil slick is
Express the area as a function of time using substitution.

13. 2.6 Evaluating Composite Functions
Example
Given find (a) and (b)
Solution
(a)
(b)

14. 2.6 Finding Composite Functions
Let and Find (a) and (b)
Solution
(a)
(b)
Note:

15. 2.6 Finding Functions that Form a Composite Function
Suppose that Find f and g so that
Solution
Note the repeated quantity
Let
Note that there are other pairs of f and g that also work.

16. 2.6 Application of Composite Functions
Finding and Analyzing Cost, Revenue, and Profit
Suppose that a businessman invests $1500 as his fixed cost in a new
venture that produces and sells a device that makes programming a
VCR easier. Each device costs $100 to manufacture.
Write a linear cost function with x equal to the quantity produced.
Find the revenue function if each device sells for $125.
Give the profit function for the item.
How many items must be sold before the company makes a profit?
Support the result with a graphing calculator.

17. 2.6 Application of Composite Functions
Solution
Using the slope-intercept form of a line, let
Revenue is price  quantity, so
Profit = Revenue – Cost
Profit must be greater than zero

18. 2.6 Application of Composite Functions
(e) Let
Figure 71a pg 2-162

19. 2.6 Applying a Difference of Functions
Example The surface area of a sphere S with radius r is
S = 4 r2.
Find S(r) that describes the surface area gained when r increases by 2 inches.
Determine the amount of extra material needed to manufacture a ball of radius 22 inches as compared to a ball of radius 20 inches.