By Hari Srivastava, H. L. Manocha
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Extra info for A treatise on generating functions
What is the y-intercept? 2 Ϫ2 41. For what values of x is f (x) > 0? 1 Ϫ2 Ϫ2 4 x 52. On what interval(s) is the function f constant? 53. On what interval(s) is the function f nonincreasing? 54. On what interval(s) is the function f nondecreasing? 13 14 Chapter P • Preparing for Calculus In Problems 55–60, answer the questions about the function 69. 70. x +2 . g(x) = x −6 y 55. What is the domain of g? 1 58. If g(x) = 2, what is x? What is(are) the corresponding point(s) on the graph of g? (1, 0) Ϫ2 57.
If x is in the domain of a function f , we say that f is defined at x, or f (x) exists. If x is not in the domain of f , we say that f is not defined at x, or f (x) does not exist. The domain of a function is expressed using inequalities, interval notation, set notation, or words, whichever is most convenient. Notice the various ways the domain was expressed in the solution to Example 3. NOW WORK Problem 17. 3 Identify the Graph of a Function In applications, often a graph reveals the relationship between two variables more clearly than an equation.
For example, in Example 3, (g ◦ f )(x) = g( f (x)) = 4 = f (x) − 1 4 4(x + 2) 4(x + 2) =− = 1 1 − (x + 2) x +1 −1 x +2 Functions f and g for which f ◦ g = g ◦ f will be discussed in the next section. Some techniques in calculus require us to “decompose” a composite function. √ For example, √ the function H (x) = x + 1 is the composition f ◦ g of the functions f (x) = x and g(x) = x + 1. EXAMPLE 4 Decomposing a Composite Function Find functions f and g so that f ◦ g = F when: √ 1 (a) F(x) = (b) F(x) = (x 3 − 4x − 1)100 (c) F(t) = 2 − t x +1 1 Solution (a) If we let f (x) = and g(x) = x + 1, then x 1 1 ( f ◦ g)(x) = f (g(x)) = = = F(x) g(x) x +1 (b) If we let f (x) = x 100 and g(x) = x 3 − 4x − 1, then ( f ◦ g)(x) = f (g(x)) = f (x 3 − 4x − 1) = (x 3 − 4x − 1)100 = F(x) √ (c) If we let f (t) = t and g(t) = 2 − t, then √ ( f ◦ g)(t) = f (g(t)) = f (2 − t) = 2 − t = F(t) ■ Although the functions f and g chosen in Example 4 are not unique, there is usually a “natural” selection for f and g that first comes to mind.
A treatise on generating functions by Hari Srivastava, H. L. Manocha