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

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

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

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

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

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

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

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

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

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.

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.

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

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

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.

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.

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

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

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.