Composition of Functions

Transcription

1 Math 120 Intermediate Algebra Sec 9.1: Composite and Inverse Functions Composition of Functions The composite function f g, the composition of f and g, is defined as (f g)(x) = f(g(x)). Recall that a function f can be thought of as a rule. Thus, f g is a combination of the rules f and g, where the rule determined by g is applied first. Ex 1 Let f(x) = 3x 2 + 4, g(x) = 4x 1, h(x) = x + 3, and j(x) = 13. Find the following: a) Do (f g)(1) b) Do (f g)(x) c) (g f)(x) x d) Do (g f)(1) e) Do (h j)(x) f) Do (j h)(x) g) Let a(x) = 2 and b(x) = 3x 7. Find (a b)(x). 3 Ex 2 Find f(x) and g(x) such that h(x) = (f g)(x). (Answers may vary.) h(x) = 4x 5 Page 1 of 16

2 Recall A function f is a relation (a set of ordered pairs) in which every x-coordinate corresponds to exactly one y-coordinate. 1-1 Function 1) A function f is one-to-one (1-1) if every y-coordinate corresponds to every x-coordinate. (Note that every x-coordinate will correspond to exactly one y-coordinate since f is a function.) 2) A function f is one-to-one (1-1) if different inputs (x-values) corresponds to different outputs (y-values). That is, if a and b are in the domain of f, a b, then f(a) f(b). Inverse Relation Let R be a relation. For every (x, y) in the relation R, reverse the values to obtain (y, x). The collection of all such (y, x) s forms the inverse relation of R, R 1. Note: If f is a 1-1 function, its inverse is also a function. If f is not 1-1, its inverse is not a function. Defn The inverse of the function f, f 1 (read f-inverse ), is defined as f 1 (y) = x, where f(x) = y. Caution: f 1 does NOT mean f raised to the 1 power. f is NOT a variable and 1 is NOT an exponent. Examples (relations & inverses, functions & inverses, 1-1 notation with ordered pairs R 1 = { } R 2 = { } R 3 = { } R 1 1 = { } R 1 2 = { } R 1 3 = { } Observation: If R is a 1-1 function, then. f(x) = x 2; f 1 (x) = f(x) = 4x; f 1 (x) = f(x) = x 3 ; f 1 (x) = f(x) = 3x 2; f 1 (x) = Inverse Notation f 1 f(3) = 4 means Page 2 of 16

3 The Horizontal Line Test If it is impossible to draw a horizontal line that intersects a function s graph more than once, then the function is one-to-one. For every one-to-one function, an inverse function exists. Ex 3 Determine whether the function is 1-1. a) f(x) = x + 5 b) f(x) = 3 x 2 c) f(x) = x 3 d) f(x) = 1 x e) f(x) = x f) f(x) = x x 2, x 4, x 6, vs x 3, x 5, x 7, Inverses of Functions Handout pg 59 and 60 (Do all problems) Finding the Inverse First make sure that f is ) Replace f(x) with y. 2) Interchange x and y. (This gives the inverse function.) 3) Solve for y. 4) Replace y with f 1 (x). (This is inverse function notation.) Composition and Inverses If f is a 1-1 function, then f 1 is the unique function for which (f 1 f)(x) = f 1 (f(x)) = x for all x in domf and (f f 1 )(x) = f(f 1 (x)) = x for all x in domf 1. Inverses of Functions Handout pg 61 and 62 (Do select problems) Ex 4 Given the graph of f below, graph f 1. Plot all pertinent points. Page 3 of 16

4 Sec 9.2: Exponential Functions Defn The function f(x) = b x, where b > 0 and b 1, is called the exponential function, base b. Note: x is any real number. That is, the domain of f is (, ). What is the range? f(x) = b x = log b x Domain Range Graphs of Exponent Functions Handout pg 63 and 64 (Do 1, 2, 3 or 4, 6-8, 10, 12, discuss 11 and shifting for all) Ex 5 Let f(x) = ( 1 2 )x + 1. Find the domain and range of f(x), f(x) 4, and f(x 4). What happens to the domain, range, and asymptote if we shift y = 2 x up, down, left, and right? Page 4 of 16

5 Sec 9.3: Logarithmic Functions Meaning of log b x = y For x > 0 and b > 0, b 1, log b x is the exponent to which b must be raised to get x. Thus, log b x = y means b y = x or equivalently, log b x is the unique exponent for which b log b x = x. The logarithmic function f(x) = log b x is the inverse of the exponential function g(x) = b x. What is the domain and range? f(x) = b x f(x) = log b x Domain Range Ex 6 Simplify. 1 a) log b) log 2 8 c) log 5 5 d) log 9 1 e) log 3 3 f) log g) log h) log Page 5 of 16

6 i) log 4 32 j) log 16 8 k) 6 log 6 13 What are we really finding? We are really finding the, which is the answer to a log expression. Ex 7 Rewrite each of the following as an equivalent exponential equation. Do not solve. a) b) c) d) y = log 8 10 log 6 36 = 2 log b n = 23 log e Ex 8 Rewrite each of the following as an equivalent logarithmic equation. Do not solve. Watch notation. a) b) c) d) 2 5 = = 2 6 1/3 3 = 6 e Ex 9 Solve. a) log 4 x = 3 b) log 5 25 = x c) log x 12 = 1 Page 6 of 16

7 d) log x 9 = 1 2 e) log 2 x = 1 f) log 8 x = 2 3 g) 3 log 64 x = 2 h) 2 log x 12 = 1 i) 8 x = 12 Graphs of Logarithm Functions Handout pg 65 and 66 (Do all problems) Writing Exponent and Log Functions Handout pg 69 and 70 (Do a few problems) Ex 10 Let f(x) = log 3 (x 2). Find the domain and range of f(x), f(x) 4, f(x 4), and f(x + 3). Few quick sketches of base functions first. Address reflections. Page 7 of 16

8 x y = 3 x -3 1/27-2 1/9-1 1/ Sec 9.4: Properties of Logarithmic Functions Sec 9.5: Common and Natural Logarithms Sec 9.6: Solving Exponential and Logarithmic Equations Sec 9.7: Applications of Exponential and Logarithmic Functions [Sec 9.3] Warm Up: Graph y = log 3 x and y = log 1/3 x. x x x [Sec 9.4 & 9.6] Log Rules Let M, N, and b be postive numbers with b 1. Let p be any real number. 1. log b (MN) = log b M + log b N 2. log b M p = p log b M Log of a the Base to an Exponent For any base b, log b b k = k. 3. log b ( M N ) = log b M log b N Ex 11 Express as an equivalent expression that is a sum or difference of logs. a) (b) c) (d) 29 log 2 (16 32) log t (3ab) log 3 13 log log 2 32 log 3 29 log 3 13 log a y x Ex 12 Express as an equivalent expression that is a product. a) b) log b t 5 log 10 y Page 8 of 16

9 Ex 13 Express as a sum/difference of logs. 4 log b x 8 y 3 Ex 14 Express as an equivalent expression that is a single log. Simplify if possible. a) DO b) c) DO 2 log b m + 1 log 2 b n a log a x a ax log a (2x + 10) log a (x 2 25) log a a x Ex 15 Given log b 3 = and log b 5 = If possible, use properties of logs to calculate values for each. a) PP b) c) d) 5 1 log b log 3 b 3 log b 25 log b [Sec 9.5 & 9.6] Common Log: log x means log 10 x Natural Log: ln x means log e x The number e is an irrational number. e Change-of-Base Formula log b M = log a M log a b Note: e is an irrational constant like π. Since e > 1, f(x) = e x is an exponential function (whose graph increases). The inverse of f(x) = e x is f 1 (x) = log e x = ln x. Page 9 of 16

10 Ex 16 Simplify. log p p 5 log e e a Rules e ln 8 ln 3x2 e The Principle of Exponential Equality For any real number b, where b 1,0, or 1, b x = b y is equivalent to x = y The Principle of Logarithmic Equality For any log base b, and for x, y > 0, x = y is equivalent to log b x = log b y Ex 17 Solve. a) 2 3x = 64 b) 9 x = 27 c) 9 x 1 = 27 d) 5 x+2 = 15 e) e t = 20 f) 4 + 5e x = 9 g) 6 x = 12 Ex 18 Solve. Must check for domain problems when solving log equations! a) log 2 x = 6 b) log 5 x = 3 c) ln(3x) = 2 Page 10 of 16

11 d) ln(4x 2) = 3 e) 3 ln x = 3 Ex 19 Solve each equation. a) b) c) log(x + 9) + log x = 1 log 6 (x + 3) + log 6 (x + 2) = log 6 20 log(x + 1) log x = 0 d) e) ln(x 6) + ln(x + 3) = ln 22 log 3 (x 4) + log 3 (x + 4) = 2 Page 11 of 16

12 f) g) log 4 (x + 6) log 4 x = 2 log 12 (x + 5) log 12 (x 4) = log 12 3 h) i) j) 10 ln e x2 = ln(x ) = ln(2x 2 9) log(2x + 3) + log(4x 1) = 2 log 3 Provide examples functions and their inverses. f(x) = log 3 x f(x) = x 3 Ex 20 Sketch the graph and state the domain and range of each function. (a) (b) (c) f(x) = e x f(x) = e x + 3 f(x) = e x+3 Page 12 of 16

13 (d) (e) (f) f(x) = e x and g(x) = ln x g(x) = ln x + 3 g(x) = e x (g) g(x) = ln(x + 3) (h) g(x) = 3 ln x Ex 21 Find the domain and range of each function. Use interval notation. a) f(x) = 3 x b) f(x) = 3 x+2 c) f(x) = 3 x + 2 domf = domf = domf = rng f = rng f = rng f = Page 13 of 16

14 d) f(x) = log 3 x e) f(x) = log 3 (x + 2) f) f(x) = log 3 x + 2 domf = domf = domf = rng f = rng f = rng f = What can be said about the domain and range of f and f 1? [Sec 9.7] Interest Compounded n Times a Year A = P (1 + r n )nt t = number of years r = interest rate P = amount of money invested A = balance after t yrs The model A = P (1 + r n )nt is used when the amount of money is compounded n times a year. Interest Compounded Continuously P(t) = P 0 e rt t = number of years r = interest rate P 0 = amount of money invested P = balance after t yrs The exponential growth model P(t) = P 0 e rt is used when the amount of money is compounded continuously (interest is computed every instant ). Compound Interest Practice Handout pg 71 and 72 (Do select problems) Additional Practice with Exponential and Log Equations (9.6) 1) 2) 3) 3 x = x 2 = x+1 = 8 Page 14 of 16

15 4) 5) 6) 5 + 6e 2x = x = 81 2x 3 3 x2 +2x = 1 3 7) 8) 9) 3 2x 3 x2 = x 4 4x+1 = x = 25 4x ) 11) 12) log 2 (x + 2) = 3 + log 2 (x 1) log 3x = log 2 + log(x + 5) log 6 (x + 1) + log 6 (x + 3) = log ) 14) 15) ln(x 8) + ln(x + 4) = ln 85 log(x + 3) log(x 2) = log 2 log x (log 3 9) = 2 16) 17) 18) 19) log 2 x 2 9 = 1 log 6 x = 3 2 log x = (log x) 2 log x (log x) = 1 Page 15 of 16

16 Sec 10.1: Conic Sections: Parabolas and Circles Defn A circle is a set of points in a plane that are a fixed distance r, called the, from a fixed point (h, k), called the. Equation of a Circle (Standard Form) Center: (h, k) (x h) 2 + (y k) 2 = r 2 Radius: r Ex 1 Find the center and radius of each circle. Graph part a). a) (x 2) 2 + (y + 3) 2 = 100 b) x 2 + y 2 + 6x 4y 15 = 0 C(, ) r = C(, ) r = Page 16 of 16