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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Warm Up Solve for x in terms of y. 1. 2. 3. 4. y = 2ln x

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. one-to-one function Vocabulary

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. You learned that the inverse of a function f(x) “undoes” f(x). Its graph is a reflection across line y = x. The inverse may or not be a function. Recall that the vertical-line test can help you determine whether a relation is a function. Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Use the horizontal-line test to determine whether the inverse of the blue relation is a function. Example 1 The inverse is a function because no horizontal line passes through two points on the graph.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Use the horizontal-line test to determine whether the inverse of the red relation is a function. Example 2 The inverse is a not a function because a horizontal line passes through more than one point on the graph.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Use the horizontal-line test to determine whether the inverse of each relation is a function. The inverse is a function because no horizontal line passes through two points on the graph. Example 3

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. To write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 4 Find the inverse of. Determine whether it is a function, and state its domain and range. Rewrite the function using y instead of f(x). Step 1 Find the inverse. Simplify. Switch x and y in the equation. Cube both sides. Isolate y.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 4 Continued The inverse is a function,. B/c? The domain of the inverse is the range of f(x):{x|x R }. The range is the domain of f(x):{y|y R }. Check Graph both relations to see that they are symmetric about y = x.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Rewrite the function using y instead of f(x). Step 1 Find the inverse. Take the cube root of both sides. Switch x and y in the equation. Add 2 to both sides of the equation. Simplify. Example 5 y = x 3 – 2 x = y 3 – 2 x + 2 = y 3 3 x + 2 = y 3 3 3 Find the inverse of f(x) = x 3 – 2. Determine whether it is a function, and state its domain and range.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. The domain of the inverse is the range of f(x): R. The range is the domain of f(x): R. Check Graph both relations to see that they are symmetric about y = x. Example 5 Continued Because the inverse is a function,.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value. You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Determine by composition whether each pair of functions are inverses. Example 6 Find the composition f(g(x)). f(g(x)) = 3( x + 1) – 1 1 3 Use the Distributive Property. Simplify. f(x) = 3x – 1 and g(x) = x + 1 1 3 Substitute x + 1 for x in f. 1 3 = (x + 3) – 1 = x + 2

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)). Example 6 Continued Check The graphs are not symmetric about the line y = x.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 7 Find the compositions f(g(x)) and g(f (x)). For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x 1 x – 1 Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses. = x = (x – 1) + 1 = x

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 7 Continued Check The graphs are symmetric about the line y = x for all x but 0 and 1.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Independent Practice Due Tomorrow at Beginning of Class p. 453-454 # 9-21, 25- 35 odd

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Lesson Quiz: Part I A: yes; B: no 1. Use the horizontal-line test to determine whether the inverse of each relation is a function.

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Lesson Quiz: Part II D: {x|x ≥ 4}; R: {all Real Numbers} 2. Find the inverse f(x) = x 2 – 4. Determine whether it is a function, and state its domain and range. not a function

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5.4 Functions and Their Inverses CC.9-12.F.BF.1c (+) Compose functions. CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9- 12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 3. Determine by composition whether f(x) = 3(x – 1) 2 and g(x) = +1 are inverses for x ≥ 0. Lesson Quiz: Part III yes

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Example 1A: Using the Horizontal-Line Test

Example 1A: Using the Horizontal-Line Test

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