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6-1 Square Root Functions as Inverses TEKS FOCUS VOCABULARY ĚSquare root function – A square root TEKS (2)(C) Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range. function is the inverse of a quadratic function with a restricted domain. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. ĚApply – use knowledge or information for a specific purpose. such as solving a problem Additional TEKS (1)(D), (2)(B), (2)(D) ESSENTIAL UNDERSTANDING The domain of a quadratic function can be restricted so that its inverse is a square root function. Key Concept Inverse of a Quadratic Function A horizontal line can intersect the graph of f (x) = x2 in two points—where f ( -2) = f (2), for example. 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. y f(x) x2 ⫺4 ⫺2 2 O 2 f1(x) x x 2 ⫺2 x O y 2 However, you can restrict the domain of f so that the inverse of the restricted function is a function. y y 2 2 f(x) x2 , x ⱖ 0 ⫺2 230 Lesson 6-1 O Square Root Functions as Inverses x 2 O f1(x) x 2 4 x Problem 1 P TEKS Process Standard (1)(D) Finding an Inverse Consider the function f (x) = x2 . A Write the inverse of the function. First, rewrite the equation as y = x2 . Since the graph of an inverse is a reflection across the line y = x, you can switch x and y in the equation to get an equation for the inverse. y = x2 x = y2 {2x = y Switch x and y. Find the square root of each side to solve for y. Using f -1 notation, this equation can also be written as f -1(x) = {2x. B Analyze and describe the relationship between the function and its inverse, including restrictions on domain and range. What will the graph of the inverse look like? Imagine folding the graph of y = x 2 along the line y = x. You will get a parabola in the same shape as y = x 2 , but opening to the right. The function and its inverse represent inverse operations. The domain of y = x2 is all real numbers. Since x2 must be nonnegative, the range is y Ú 0. Square root functions are defined for nonnegative real numbers, so the domain of y = {2x is x Ú 0, and the range is all real numbers. C Graph the inverse. To graph y = {2x, first graph y = 2x, then graph y = - 2x. The graphs of f (x) and f -1(x) are parabolas, each a reflection of the other across the line y = x. The graph of f -1(x) shows that it is not a function. y 2 x O -2 5 y=± x D Restrict the domain of f so that the inverse of f (x) = x2 is a function. What is the equation of the inverse? When the domain of f (x) = x2 is restricted to x Ú 0, the range is y Ú 0. The inverse of the restricted function is f -1(x) = 2x with a domain of x Ú 0. The range is y Ú 0. The graph shows that the inverse is a function because it passes the vertical line test. y y 2 2 f(x) x2 , x ⱖ 0 ⫺2 O x 2 O f1(x) x 2 4 x PearsonTEXAS.com 231 Problem 2 P Graphing a Function and Its Inverse Consider the function f (x) = x2 + 3 with domain x # 0, and its inverse, f −1(x) = 2x − 3. 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. A Analyze and describe the relationship between the function and its inverse. Since f (x) is a quadratic function with an appropriately restricted domain, the inverse is a square root function. The graph of f -1(x) is a reflection of the graph of f (x) across the line y = x. B What are the domain and range of f (x) and f −1(x)? Write your answers as inequalities. The domain of f is restricted to x Ú 0. The range of f is y Ú 3. The domain of f -1 is x Ú 3, and the range of f -1 is y Ú 0. C Graph f (x) and f −1(x). + 3, translate the graph of To graph f (x) = f (x) = x2 up three units and only graph x-values greater than or equal to zero. To graph the inverse, reflect the graph of f (x) across the line y = x. x2 Problem bl 3 f(x) 8 6 4 2 O TEKS Process Standard (1)(A) The function d = 4.9t 2 represents the distance d, in meters, that an object falls in t seconds due to Earth’s gravity. 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 2 d t 2 = 4.9 232 Lesson 6-1 x 5 f−1(x) = x− 3, x ≥ 3 -2 Finding the Inverse of a Formula Why shouldn’t you interchange the variables? Interchanging the variables leads to a false relationship between distance and time. f(x) = x2 + 3, x≥0 d t = 5 4.9 24 = 5 4.9 ≈ 2.2 Solve for t. Do not switch the variables. Time must be nonnegative. Substitute 24 for d. Use a calculator. It I will take about 2.2 seconds for the diver to reach the t water. Square Root Functions as Inverses 24 meters Problem 4 P Composing Functions How does composition show that two functions are inverses? The composition of a function and its inverse is the identity function. So as long as you choose an x value within the domain, if the functions are inverses, the result of the composition should be the original x value. U composition to show that f (x) = x2 + 7 with domain x # 0 and Use g(x) = 1x − 7 are inverse functions. g In I general, if ( g ∘ f )(x) = x and ( f ∘ g)(x) = x for x in the domains of f and g, respectively, then f and g are inverse functions. r Check: C ( f ∘ g)(x) = f (g(x)) (g ∘ f )(x) = g( f (x)) ( = f 2x - 7 ( = 2x - 7 ) )2 + 7 = g(x2 + 7) = 2(x2 + 7) - 7 =x-7+7 = 2x2 =x = x = x (since x Ú 0) NLINE HO ME RK O So f (x) and g(x) are inverse functions. S WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. For Exercises 1–2, write the inverse of each function. 1. Write the inverse of f (x) = 5x - 4. For additional support when completing your homework, go to PearsonTEXAS.com. 2. Write the inverse of f (x) = (x + 8)2 . Give the domain and range of the function and its inverse. 3. Write and graph the inverse of y = - x2 + 2. 4. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Find the inverse of y = 4x2 . Graph both the function and its inverse. Explain how the equations and graphs show the relationship between the function and its inverse. 5. Graph the function f (x) = 13 x - 2 and its inverse, f -1(x) = 3x + 6. 6. Graph the function f (x) = (x + 2)2 and its inverse, f -1(x) = { 2x - 2. How would you 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 (x) = - x2 + 4 and its inverse, f -1(x) = {24 - x? Write your 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. Analyze Mathematical Relationships (1)(F) The area of a circle is given by the equation A = pr 2 , where r is the radius of the circle. The inverse function is A r = 5p . Analyze and describe the relationship between the functions. Write the domain and range of both functions as inequalities. PearsonTEXAS.com 233 Use composition to show that f and g are inverse functions. 9. f (x) = 12 x + 3 and g(x) = 2x - 6 x 10. f (x) = 5 5 and g(x) = 5x2, x 7 0 1-x 11. f (x) = 1 - 2x2, x Ú 0 and g(x) = 5 2 12. Nina belongs to a gym that charges $35 per month plus a $95 enrollment fee. She has found that the equation f (x) = 35x + 95 gives the total amount she has paid for x months. Find the inverse of this function. Then use composition to check your answer. 13. The formula for converting from Celsius to Fahrenheit temperatures is F = 95C + 32. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the Celsius temperature that corresponds to 25°F. 14. V = 43pr 3 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 ft3. 15. Apply Mathematics (1)(A) The velocity 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 (x) = 12gx models the velocity v in feet per second where g, the acceleration due to gravity, is about 32 ft>s2 and x 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 20 ft/s? 16. Let f (x) = 3x2 - 4 and g (x) = x - 2. Calculate ( f ∘ g -1)(x) for x = -3. 17. Explain Mathematical Ideas (1)(G) Explain how you can find the range of the inverse of f (x) = 1x - 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 (x) = - 1x 19. f (x) = 1x + 3 20. f (x) = 1-x + 3 21. f (x) = 1x + 2 x2 23. f (x) = 12 22. f (x) = 2 24. f (x) = (x 26. f (x) = 234 Lesson 6-1 x 4)2 1 (x + 1)2 Square Root Functions as Inverses 25. f (x) = (7 - x)2 27. f (x) = 4 - 21x 28. a. Display Mathematical Ideas (1)(G) Copy the mapping diagram at the right. Complete it by 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. Relation r Domain Range b. Repeat part (a) so that r is not a function and r -1 is a function. 29. Explain 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? Explain. 30. Apply Mathematics (1)(A) Write a function that gives the length of the hypotenuse 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 hypotenuse of 6 in. 3 31. For the function f (x) = 1 2x, find f -1(x). Then determine the value of x when f (x) = 16. TEXAS Test Practice T 32. Which pair of words makes this sentence FALSE? The product of two ____(I)____ numbers is always a(n) ____(II)____ number. A. (I) complex; (II) complex B. (I) real; (II) complex C. (I) rational; (II) real D. (I) imaginary; (II) imaginary 33. If f (x) = x + 1 and g (x) = x2 - 3x - 4, what is ( f ∘ g)(x)? F. x2 - 3x - 3 G. x2 - x - 6 H. x2 - x J. x2 - x - 3 2 3 ( ) 2 34. What is the simplified form of a3b4 ? 4 9 A. a9b16 4 3 B. a3b2 C. ab 17 D. (ab) 6 35. Let f (x) = (x + 1)2 - 2. Find the x- and y-intercepts of f (x) and the inverse of f (x). Is the inverse a function? PearsonTEXAS.com 235