Chapter 2: Analysis of Graphs of Functions

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 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

3. 2.6 Operations and Composition 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. • The domains 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 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 (a) (b) (c)

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

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 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 thatFind 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 businesswoman invests $1500 as her fixed cost in a new venture that produces and sells a device for satellite radio. Each device costs $100 to manufacture. • Write a cost function for the product if x represents the number of devices produced. Assume that the function is linear. • Find the revenue function if each device in part (a) sells for $125. • Give the profit function for the item in parts (a) and (b). • How many items must be produced and sold before the company makes a profit?

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 Applying a Difference of Functions Example The formula for the surface area S of a sphere is S = 4r2, where r is the radius of the sphere. • Construct a model D(r)that describes the amount of surface area gained when r is increased by 2 inches. • Determine the amount of extra material needed to manufacture a ball of radius 22 inches compared to a ball of radius 20 inches.