MATH 150 TOPIC 3 FUNCTIONS: COMPOSITION AND DIFFERENCE QUOTIENTS

Transcription

1 Mat 50 T3-Functions and Difference Quotients Review Page MATH 50 TOPIC 3 FUNCTIONS: COMPOSITION AND DIFFERENCE QUOTIENTS I. Composition of Functions II. Difference Quotients Practice Problems

2 Mat 50 T3-Functions and Difference Quotients Review Page 2 I. Composition of Functions HAPPY BIRTHDAY... OK so in all likeliood today isn t your birtday. Let s assume it is and you ve been given a present wrapped in a large box. You open te box and surprise!!!, inside is anoter box. Undaunted you open it and discover yet anoter box. Tis may appear to be a lesson in accepting disappointment. Unbeknownst to you, toug, you are experiencing composite function beavior. HUH?!?! Let us explain. Te composite function is like aving one function contained inside anoter. Wen you see 2x, you probably tink of it as just anoter function, but it s someting more. It s a composite (one function inside anoter). Here s wy. Suppose f(x) x and g(x) 2x (tink of f( ) ( ) as te larger box). Now place 2x inside. Matematically tis is written as f(g(x)) and we call it f composite g. To form a composite, try te following: f(g(x)) f( ) ( ). Now fill in te blanks wit te g(x) representation and you get f(g(x)) f(2x ) 2x. NOTE: just like f(3) means to input 3 into f, f(g(x)) means to input g into f. Unlike boxes, any function can be placed inside anoter. From above, g(x) 2x. Tis really means g( ) 2( ). If f(x) x,weave g(f(x)) g( x)2( x) 2 x. Alternative notation for composite functions: g(f(x)) g f, f(g(x)) f g.

3 Mat 50 T3-Functions and Difference Quotients Review Page 3 Let s try a different practice drill. Suppose (x) sin(2x). Can you state functions for f and g so tat f(g(x))? Solution: inner function is 2x, wicisg, outer function is sin x sin( ), wic is f. Ceck: f(g(x)) f( ) sin( ). Now insert 2x and get f(g(x)) f(2x) sin2x. NOTE: Tis is te most obvious coice for f and g but not te only one. Let us try f(x) sin(x +)andg(x) 2x. Ceck: f(g(x)) f( ) sin[( ) + ], so f(g(x)) f(2x ) sin[(2x ) + ] sin 2x. Wat s so special about composite functions? One day in Calc I your instructor will trow a birtday party for te entire class, and is present to you will be called te CHAIN RULE. Exercise: Practice Problem 3.. II. Difference Quotients A. Introduction In algebra, rate of cange is introduced in its most basic form by finding te slope of a line using m y 2 y x 2 x. Tis formula can also be called a Difference Quotient. In calculus, rates of cange (bot average and instantaneous) are found using function forms of difference quotients. Let s start wit a few examples of te algebra in difference quotients.

4 Mat 50 T3-Functions and Difference Quotients Review Page 4 Example: Let g(x) x 2 +. Evaluate and simplify te difference g(x) g(2) quotient, x 2. x 2 Solution: Using g(x) x 2 +andg(2) 5, g(x) g(2) x 2 (x2 +) 5 x 2 x2 4 x 2 x +2. Example: Let g(x) x 2. Evaluate and simplify te difference g(x +) g(x) quotient, 0[represents te cange in x.] Remember: g( ) g(x +) g(x) ( ) 2. (x +) 2 x 2 (x 2) (x + 2) (x + 2)(x 2) (x + 2)(x 2)() (x + 2)(x 2)

5 Mat 50 T3-Functions and Difference Quotients Review Page 5 B. Application How does a difference quotient measure rate of cange (or slope)? Suppose we start wit a function f(x) and draw a line (called a secant line) connecting two of its points. f(x) (x +, f(x + )) (x, f(x)) x x + m y 2 y x 2 x y f(x + ) f(x) (x + ) x f(x + ) f(x) Notice tat te slope is represented by a difference quotient and is referred to as te average rate of cange. Now suppose te distance between tese two points is sortened by decreasing te size of. y f(x) tangent line x Conclusion: Te two points get so close togeter tey almost concide. Te resulting line is now a tangent line wose slope measures te

6 Mat 50 T3-Functions and Difference Quotients Review Page 6 instantaneous rate of cange. You will come to know tis slope by te term derivative. PRACTICE PROBLEMS for Topic 3 Functions and Difference Quotients 3. a) Suppose f(x) x 2 2x and g(x). Form te following x compositions: i) f(g(x)) ii) g(f(x)) iii) f(f(x)) b) Suppose f(x) sinx and g(x) x. Form te following compositions: i) f(g(x)) ii) g(f(x)) c) Suppose y sin 3x. Select functions for f, g, and so tat y f(g())(x). Ceck your results. Return to Review Topic Answers 3.2 Find te rate of cange (slope) of te line containing te given points. a) (2, 5) and (4,) b) (3, 4) and (3, ) c) (x, f(x)) and (x +, f(x + )) Answers 3.3 Evaluate and simplify te difference quotient f(x + ) f(x), 0, wen: a) f(x) x 2 3x; b) f(x) x 2 (Rationalize te numerator). Answers f(x +) f(x) 3.4 Evaluate and simplify te difference quotient, 0, wen f(x) 2+x. Answer

7 Mat 50 T3-Functions and Difference Quotients Review Answers Page 7 ANSWERS to PRACTICE PROBLEMS (Topic 3 Functions and Difference Quotients) 3. a) i) ii) ( ) 2 2 x x 2 2x ( ) x x 2 2 x iii) (x 2 2x) 2 2(x 2 2x) (x 2 2x)(x 2 2x 2) b) i) sin x ii) sin x c) inner: (x) 3x middle: g(x) sinx outer: f(x) x CHECK: f(g())(x) f(g(3x)) f(sin 3x) sin 3x g f Return to Problem 3.2 a) m 3 b) m is undefined c) m f(x + ) f(x) Return to Problem

8 Mat 50 T3-Functions and Difference Quotients Review Answers Page a) b) (x + ) 2 3(x + ) (x 2 3x) 2x + 2 3x 2x + 3 x + 2 x 2 x + 2+ x 2 x + 2+ x 2 ( x + 2) 2 ( x 2) 2 ( x + 2+ x 2) x + 2 (x 2) ( x + 2+ x 2) x + 2+ x 2 Return to Problem (x +) 2+x 2+x (2 + x +) (2 + x +)(2 + x) (2 + x +)(2 + x) Return to Problem Beginning of Topic 50 Review Topics Skills Assessment