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7-7 Inverse Relations & Functions

7-7 Inverse Relations & Functions. M11.D.1.1.3: Identify the domain, range, or inverse of a relation. Objectives. The Inverse of a Function. x –1 0 1 2 y –2 –1 –1 –2. Interchange the x and y columns. Inverse of Relation m. x –2 –1 –1 –2 y –1 0 1 2.

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7-7 Inverse Relations & Functions

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  1. 7-7 Inverse Relations & Functions M11.D.1.1.3: Identify the domain, range, or inverse of a relation

  2. Objectives The Inverse of a Function

  3. x –1 0 1 2 y –2 –1 –1 –2 Interchange the x and y columns. Inverse of Relation m x –2 –1 –1 –2 y –1 0 1 2 Finding the Inverse of a Relation a. Find the inverse of relation m. Relation m

  4. Reversing theOrdered Pairs Relation m Inverse of m Continued b. Graph m and its inverse on the same graph.

  5. ± x + 2 = yFind the square root of each side. Interchanging x and y Find the inverse of y = x2 – 2. y = x2 – 2 x = y2 – 2 Interchange x and y. x + 2 = y2Solve for y.

  6. The graph of y = –x2 – 2 is a parabola that opens downward with vertex (0, –2). The reflection of the parabola in the line x = y is the graph of the inverse. Graphing a Relation and Its Inverse Graph y = –x2 – 2 and its inverse. You can also find points on the graph of the inverse by reversing the coordinates of points on y = –x2 – 2.

  7. ƒ(x) = 2x + 2 Rewrite the equation using y. y = 2x + 2 x2 – 2 2 x2 – 2 2 y = Interchange x and y. x2 = 2y + 2 Square both sides. x = 2y + 2 Solve for y. So, ƒ –1(x) = . Finding an Inverse Function Consider the function ƒ(x) = 2x + 2 . a. Find the domain and range of ƒ. Since the radicand cannot be negative, the domain is the set of numbers greater than or equal to –1. Since the principal square root is nonnegative, the range is the set of nonnegative numbers. b. Find ƒ –1

  8. > > – – x2 – 2 2 Since x2 0, –1. Thus the range of ƒ–1 is the set of numbers greater than or equal to –1. Note that the range of ƒ–1 is the same as the domain of ƒ. Continued (continued) c. Find the domain and range of ƒ–1. The domain of ƒ–1 equals the range of ƒ, which is the set of nonnegative numbers. d. Is ƒ–1 a function? Explain. For each x in the domain of ƒ–1, there is only one value of ƒ–1(x). So ƒ–1 is a function.

  9. Solve for t. Do not interchange variables. t2 = d 16 Quantity of time must be positive. t = d 4 1 4 t = 50 1.77 Real-World Example The function d = 16t2 models the distance d in feet that an object falls in t seconds. Find the inverse function. Use the inverse to estimate the time it takes an object to fall 50 feet. d = 16t2 The time the object falls is 1.77 seconds.

  10. Vocabulary If and are inverse functions, then and

  11. and (ƒ°ƒ–1)(– 86) = – 86. Composition of Inverse Functions 1 2 For the function ƒ(x) = x + 5, find (ƒ–1°ƒ)(652) and (ƒ°ƒ–1)(– 86). Since ƒ is a linear function, so is ƒ–1. Therefore ƒ–1 is a function. So (ƒ–1°ƒ)(652) = 652

  12. Homework p 410 #1,2,5,6,14,15,23,24,31,32

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