6-1 Square Root Functions as Inverses TEKS FOCUS TEKS ()(C) Describe and analze the relationship between a function and its inverse (quadratic and square root, logarithmic and eponential), including the restriction(s) on domain, which will restrict its range. TEKS (1)(A) Appl mathematics to problems arising in everda life, societ, and the workplace. Additional TEKS (1)(D), ()(B), ()(D) VOCABULARY Square root function A square root function is the inverse of a quadratic function with a restricted domain. Appl use knowledge or information for a specific purpose. such as solving a problem ESSENTIAL UNDERSTANDING The domain of a quadratic function can be restricted so that its inverse is a square root function. Ke Concept Inverse of a Quadratic Function A horizontal line can intersect the graph of f () = in two points where f (-) = f (), for eample. Thus, a vertical line can intersect the graph of f -1 in two points. f -1 is not a function because it fails the vertical line test. f 1 () f() 4 O O However, ou can restrict the domain of f so that the inverse of the restricted function is a function. f(), 0 O O f 1 () 4 30 Lesson 6-1 Square Root Functions as Inverses
Problem 1 TEKS Process Standard (1)(D) Finding an Inverse Consider the function f () =. A Write the inverse of the function. First, rewrite the equation as =. Since the graph of an inverse is a reflection across the line =, ou can switch and in the equation to get an equation for the inverse. = = Switch and. { = Find the square root of each side to solve for. Using f -1 notation, this equation can also be written as f -1 () = {. What will the graph of the inverse look like? Imagine folding the graph of = along the line =. You will get a parabola in the same shape as =, but opening to the right. B Analze and describe the relationship between the function and its inverse, including restrictions on domain and range. The function and its inverse represent inverse operations. The domain of = is all real numbers. Since must be nonnegative, the range is Ú 0. Square root functions are defined for nonnegative real numbers, so the domain of = { is Ú 0, and the range is all real numbers. C Graph the inverse. To graph = {, first graph =, then graph =-. The graphs of f () and f -1 () are parabolas, each a reflection of the other across the line =. The graph of f -1 () shows that it is not a function. O 5 - = ± D Restrict the domain of f so that the inverse of f () = is a function. What is the equation of the inverse? When the domain of f () = is restricted to Ú 0, the range is Ú 0. The inverse of the restricted function is f -1 () = with a domain of Ú 0. The range is Ú 0. The graph shows that the inverse is a function because it passes the vertical line test. f(), 0 O O f 1 () 4 PearsonTEXAS.com 31
Problem Graphing a Function and Its Inverse How are the domain and range of a function and its inverse related? The domain of a function is the range of its inverse, and the range of a function is the domain of its inverse. Consider the function f () = + 3 with domain # 0, and its inverse, f 1 () = 3. A Analze and describe the relationship between the function and its inverse. Since f () is a quadratic function with an appropriatel restricted domain, the inverse is a square root function. The graph of f -1 () is a reflection of the graph of f () across the line =. B What are the domain and range of f () and f 1 ()? Write our answers as inequalities. The domain of f is restricted to Ú 0. The range f() of f is Ú 3. f() = 8 + 3, The domain of f -1 0 is Ú 3, and the range of f -1 is Ú 0. 6 C Graph f () and f 1 (). To graph f () = + 3, translate the graph of f () = up three units and onl graph -values greater than or equal to zero. To graph the inverse, reflect the graph of f () across the line =. 4 - O 5 f 1 () = 3, 3 Problem 3 Finding the Inverse of a Formula TEKS Process Standard (1)(A) Wh shouldn t ou interchange the variables? Interchanging the variables leads to a false relationship between distance and time. The function d = 4.9t represents the distance d, in meters, that an object falls in t seconds due to Earth s gravit. Find the inverse of this function. How long, in seconds, does it take for the cliff diver shown to reach the water below? d = 4.9t t = d 4.9 d t = 5 4.9 4 = 5 4.9. Solve for t. Do not switch the variables. Time must be nonnegative. Substitute 4 for d. Use a calculator. It will take about. seconds for the diver to reach the water. 4 meters 3 Lesson 6-1 Square Root Functions as Inverses
Problem 4 Composing Functions How does composition show that two functions are inverses? The composition of a function and its inverse is the identit function. So as long as ou choose an value within the domain, if the functions are inverses, the result of the composition should be the original value. Use composition to show that f () = + 7 with domain # 0 and g() = 1 7 are inverse functions. In general, if ( g f )() = and ( f g)() = for in the domains of f and g, respectivel, then f and g are inverse functions. Check: ( f g)() = f (g()) (g f )() = g( f ()) = f ( - 7) = g( + 7) = ( - 7) + 7 = ( + 7) - 7 = - 7 + 7 = = = = (since Ú 0) So f () and g() are inverse functions. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For Eercises 1, write the inverse of each function. For additional support when completing our homework, go to PearsonTEXAS.com. 1. Write the inverse of f () = 5-4.. Write the inverse of f () = ( + 8). Give the domain and range of the function and its inverse. 3. Write and graph the inverse of =- +. 4. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Find the inverse of = 4. Graph both the function and its inverse. Eplain how the equations and graphs show the relationship between the function and its inverse. 5. Graph the function f () = 1 3 - and its inverse, f -1 () = 3 + 6. 6. Graph the function f () = ( + ) and its inverse, f -1 () = { -. How would ou restrict the domain of f so that its inverse is a function? What is the equation of the inverse function? 7. a. What are the domain and range of f () = - + 4 and its inverse, f -1 () = {4 -? Write our answers in interval notation. b. Restrict the domain of f so that its inverse is a function. What is the equation of the inverse function? 8. Analze Mathematical Relationships (1)(F) The area of a circle is given b the equation A = pr, where r is the radius of the circle. The inverse function is r = 5 A p. Analze and describe the relationship between the functions. Write the domain and range of both functions as inequalities. PearsonTEXAS.com 33
Use composition to show that f and g are inverse functions. 9. f () = 1 + 3 and g() = - 6 10. f () = 5 5 and g() = 5, 7 0 11. f () = 1-1 -, Ú 0 and g() = 5 1. Nina belongs to a gm that charges $35 per month plus a $95 enrollment fee. She has found that the equation f () = 35 + 95 gives the total amount she has paid for months. Find the inverse of this function. Then use composition to check our answer. 13. The formula for converting from Celsius to Fahrenheit temperatures is F = 9 5 C + 3. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the Celsius temperature that corresponds to 5 F. 14. V = 4 3 pr3 is the formula for the volume of a sphere. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the radius of a sphere that has a volume of 35,000 ft 3. 15. Appl Mathematics (1)(A) The velocit of the water that flows from an opening at the base of a tank depends on the height of water above the opening. The function v () = 1g models the velocit v in feet per second where g, the acceleration due to gravit, is about 3 ft>s and is the height in feet of the water. What is the depth of water when the flow is 40 ft/s, and when the flow is 0 ft/s? 16. Let f () = 3-4 and g () = -. Calculate ( f g -1 )() for =-3. 17. Eplain Mathematical Ideas (1)(G) Eplain how ou can find the range of the inverse of f () = 1-1 without finding the inverse itself. For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function. 18. f () =-1 19. f () = 1 + 3 0. f () = 1- + 3 1. f () = 1 +. f () = 3. f () = 1 4. f () = ( - 4) 5. f () = (7 - ) 6. f () = 1 ( + 1) 7. f () = 4-1 34 Lesson 6-1 Square Root Functions as Inverses
8. a. Displa Mathematical Ideas (1)(G) Cop the mapping diagram at the right. Complete it b writing members of the domain and range and connecting them with arrows so that r is a function and r -1 is not a function. b. Repeat part (a) so that r is not a function and r -1 is a function. 9. Eplain Mathematical Ideas (1)(G) Relation r has one element in its domain and two elements in its range. Is r a function? Is the inverse of r a function? Eplain. 30. Appl Mathematics (1)(A) Write a function that gives the length of the hpotenuse of an isosceles right triangle with side length s. Evaluate the inverse of the function to find the side length of an isosceles right triangle with a hpotenuse of 6 in. 31. For the function f () = 1 3, find f -1 (). Then determine the value of when f () = 16. Relation r Domain Range TEXAS Test Practice 3. Which pair of words makes this sentence FALSE? The product of two (I) numbers is alwas a(n) (II) number. A. (I) comple; (II) comple B. (I) real; (II) comple C. (I) rational; (II) real D. (I) imaginar; (II) imaginar 33. If f () = + 1 and g () = - 3-4, what is ( f g)()? F. - 3-3 G. - - 6 H. - J. - - 3 34. What is the simplified form of (a 3 b 3 4 )? A. a 4 9 b16 9 B. a 4 3 b 3 C. ab D. (ab) 17 6 35. Let f () = ( + 1) -. Find the - and -intercepts of f () and the inverse of f (). Is the inverse a function? PearsonTEXAS.com 35