ca_m6 - MAC 1105 Module 6 Composite Functions and Inverse...

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Unformatted text preview: MAC 1105 Module 6 Composite Functions and Inverse Functions Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. 7. Perform arithmetic operations on functions. Perform composition of functions. Calculate inverse operations. Identity one-to-one functions. Use horizontal line test to determine if a graph represents a one-to-one function. Find inverse functions symbolically. Use other representations to find inverse functions. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Composite Functions and Inverse Functions There are two major topics in this module: - Combining Functions; Composite Functions Inverse Functions Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 Five Ways of Combining Functions (g=)g ))f()g f+)fx() g=-() ( (+ xxx x ())f()g f-xx() (× xx (öf()w)¹0 fgx () = f÷ x æ ghere () = çxg ( gx(()) è()fg ø=x (g ) fo If f(x) and g(x) both exist, the sum, difference, If and both product, quotient and composition of two functions f and g are defined by Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Example of Addition of Functions Let f(x) = x2 + 2x and g(x) = 3x - 1 Find the symbolic representation for the function f + g Find and use this to evaluate (f + g)(2). and (f + g)(x) = (x2 + 2x) + (3x − 1) (f + g)(x) = x2 + 5x − 1 (f + g)(2) = 22 + 5(2) − 1 = 13 5( or (f + g)(2) = f(2) + g(2) )(2) (2) (2) = 22 + 2(2) + 3(2) − 1 = 13 13 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 Example of Subtraction of Functions Let f(x) = x2 + 2x and g(x) = 3x − 1 Find the symbolic representation for the function f − g and use this to evaluate (f − g)(2). (f − g)(x) = (x2 + 2x) − (3x − 1) 3x (f − g)(x) = x2 − x + 1 So (f − g)(2) = 22 − 2 + 1 = 3 So Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 Example of Multiplication of Functions Let f(x) = x2 + 2x and g(x) = 3x − 1 Find the symbolic representation for the function fg and use this to evaluate (fg)(2) (fg)(x) = (x2 + 2x)(3x − 1) (fg)(x) = 3x3 + 6x2 − x2 − 2x (fg)(x) = 3x3 + 5x2 − 2x So (fg)(2) = 3(2)3 +5(2)2 − 2(2) = 40 So 3(2) +5(2) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 Example of Division of Functions Let f(x) = x2 + 2x and g(x) = 3x − 1 Find the symbolic representation for the function Find and use this to evaluate So So Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 Example of Composition of Functions Let f(x) = x2 + 2x and g(x) = 3x - 1 Find the symbolic representation for the function f ° g and use this to evaluate (f ° g)(2) )(2) (f ° g)(x) = f(g(x)) = f(3x – 1) = (3x – 1)2 + 2(3x – 1) )) (f ° g)(x) = (3x – 1) ( 3x – 1) + 6x – 2 (f ° g)(x) = 9x2 – 3x – 3x + 1 + 6x – 2 (f ° g)(x) = 9x2 – 1 So (f ° g)(2) = 9(2)2 – 1 = 35 35 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 How to Evaluate Combining of Functions Numerically? Given numerical representations for f and g in the table Evaluate combinations of f and g as specified. specified. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 How to Evaluate Combining of Functions Numerically? (Cont.) (f + g)(5) = f(5) + g(5) = 8 + 6 = 14 14 (fg)(5) = f(5) • g(5) = 8 • 6 = 48 48 (f ° g)(5) = f(g(5)) = f(6) = 7 Try to work out the rest of them now. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 How to Evaluate Combining of Functions Numerically? (Cont.) Check your answers: Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 How to Evaluate Combining of Functions Graphically? Use graph of f and g below to evaluate (f + g) (1) y = f(x) (f – g) (1) (f • g) (1) (f/g) (1) (f ° g) (1) Can you identify the two functions? Try to evaluate them now. Hint: Look at the y-value when x = 1. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. y = g(x) 13 How to Evaluate Combining of Functions Graphically? y = f(x) Check your answer now. (f + g) (1) = f(1) + g(1) = 3 + 0 = 3 (f – g) (1) = f(1) – g(1) = 3 – 0 = 3 y = g(x) (fg) (1) = f(1) • g(1) = 3 • 0 = 0 (f/g) (1) is undefined, because division by 0 is undefined. (f ° g) (1) = f(g(1)) = f(0) = 2 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 Next, Let’s Look at Inverse Functions and Their Representations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 A Quick Review on Function • y = f(x) means that given an input x, there is just one corresponding output y. • Graphically, this means that the graph passes the vertical line test. • Numerically, this means that in a table of values for y = f(x) there are no x-values repeated. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 A Quick Example Given y2 = x, iis y = f(x)? That is, is y a function of s )? x? No, because if x = 4, y could be 2 or –2. No, Note that the graph fails the vertical line test. x y 4 –2 1 –1 0 0 1 1 4 2 Note that there is a value of x in the table for which Note there are two different values of y (that is, x-values -values are repeated.) are Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 What is One-to-One? • Given a function y = f(x), f is one-to-one means that given an output y there was just one input x which produced that output. • Graphically, this means that the graph passes the horizontal line test. Every horizontal line intersects the graph at most once. • Numerically, this means the there are no yvalues repeated in a table of values. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 Example • Given y = f(x) = |x|, is f one-to-one? Given |, one-to-one – No, because if y = 2, x could be 2 or – 2. No, • Note that the graph fails the horizontal line test. Note horizontal x y –2 2 –1 1 0 0 1 1 2 2• Rev.S08 (-2,2) (2,2) Note that there is a value of y in the table for Note which there are two different values of x (that is, y-values are repeated.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 What is the Definition of a One-to-One Function? A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies f(c) ≠ f(d) Example: Given y = f(x) = |x|, f is not one-to-one because –2 ≠ 2 yet | –2 | = | 2 | Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 What is an Inverse Function? f -1 is a symbol for the inverse of the function f, not to be confused with the reciprocal. If f -1(x) does NOT mean 1/ f(x), what does it mean? y = f -1(x) means that x = f(y) Note that y = f -1(x) is pronounced “y equals f inverse of x.” Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Example of an Inverse Function Let F be Fahrenheit temperature and let C be Centigrade temperature. F = f(C) = (9/5)C + 32 C = f -1(F) = ????? • The function f multiplies an input C by 9/5 and adds 32. • To undo multiplying by 9/5 and adding 32, one should subtract 32 and divide by 9/5 So C = f -1(F) = (F – 32)/(9/5) = (5/9)(F – 32) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 Example of an Inverse Function (Cont.) F = f(C) = (9/5)C + 32 C = f -1(F) = (5/9)(F – 32) • Evaluate f(0) and interpret. f(0) = (9/5)(0) + 32 = 32 When the Centigrade temperature is 0, the Fahrenheit temperature is 32. • Evaluate f -1(32) and interpret. f -1(32) = (5/9)(32 - 32) = 0 When the Fahrenheit temperature is 32, the Centigrade temperature is 0. Note that f(0) = 32 and f -1(32) = 0 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 Graph of Functions and Their Inverses -1 • The graph of f -1 is a reflection of the graph of reflection f across the line y = x -1 Note that the domain of f equals the range of f -1 and the Note -1 range of f equals the domain of f -1 . Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 How to Find Inverse Function Symbolically? • -1 Check that f is a one-to-one function. If not, f -1 does not Check one-to-one If exist. exist. • Solve the equation y = f(x) for x, resulting in the Solve for resulting -1 -1(y) equation x = f -1 • Interchange x and y to obtain y = f -1(x) Interchange Example. Example. – – – – – – Rev.S08 Step 1 - Is this a one-to-one function? Yes. f(x) = 3x + 2 Step Step 2 - Replace f(x) with y: y = 3x + 2 Step f(x) Step 3 - Solve for x: 3x = y – 2 x: x = (y – 2)/3 2)/3 Step 4 - Interchange x and y: y = (x – 2)/3 y: -1 -1 Step 5 - Replace y with f -1(x): So f -1(x) = (x – 2)/3 Step x): http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 How to Evaluate Inverse Function Numerically? x f(x) 1 –5 2 –3 3 0 4 3 5 5 Rev.S08 -1 The function is one-to-one, so f -1 one-to-one exists. exists. -1 f -1(–5) = 1 -1 f -1(–3) = 2 -1 f -1(0) = 3 -1 f -1(3) = 4 -1 f -1(5) = 5 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 How to Evaluate Inverse Function Graphically? • The graph of f below passes the horizontal line test so f is one-toline one-toone. • Evaluate f -1(4). Evaluate -1 • Since the point (2,4) is Since on the graph of f, the the point (4,2) will be on the -1 graph of f -1 and thus -1 f -1(4) = 2 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. f(2)=4 27 What is the Formal Definition of Inverse Functions? -1 Let f be a one-to-one function. Then f -1 is the inverse function of f, iif f -1 -1 • (f -1 o f)(x) = f -1(f(x)) = x for every x in the domain )) of f -1 -1 • (f o f -1 )(x) = f(f -1(x)) = x for every x in the domain )) -1 of f -1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. 7. Perform arithmetic operations on functions. Perform composition of functions. Calculate inverse operations. Identity one-to-one functions. Use horizontal line test to determine if a graph represents a one-to-one function. Find inverse functions symbolically. Use other representations to find inverse functions. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 ...
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