Inverse Relations and Functions Lesson 7
Students will learn about finding inverse functions, solving equations for simple functions with inverses, verifying functions through composition, reading values from graphs or tables, and creating invertible functions from non-invertible ones. Key vocabulary includes inverse functions and the horizontal line test.
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Inverse Relations and Functions Unit 1 Lesson 7
Inverse Relations and Functions Students will be able to: Find inverse functions. Solve an equation of the form ? ? = c for a simple function ? that has an inverse and write an expression for the inverse. Verify by composition that one function is the inverse of another. Read values of an inverse function from a graph or a table, given that the function has an inverse. Produce an invertible function from a non-invertible function by restricting the domain.
Inverse Relations and Functions Key Vocabulary: Inverse function One-to one function Horizontal line test
Inverse Relations and Functions The inverse of a relation is a relation obtained by reversing or swapping the coordinates of each ordered pair in the relation. If the relation is described by an equation in the variables ? and ?, the equation of the inverse relation is obtained by replacing every ? in the equation with ? and every ? in the equation with ?.
Inverse Relations and Functions If ? ? represents a function of ?, the inverse of the function is represented by the symbol ? ?? . ? ? ?? ? ?.
Inverse Relations and Functions Sample Problem 1: Find the inverse of each relation given as a set of ordered pairs. a. ? ? ? ? ? ? ? ? ? ?
Inverse Relations and Functions Sample Problem 1: Find the inverse of each relation given as a set of ordered pairs. a. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Inverse Relations and Functions Sample Problem 1: Find the inverse of each relation given as a set of ordered pairs. b. ? ? ? ? ? ? ? ?? ? ?
Inverse Relations and Functions Sample Problem 1: Find the inverse of each relation given as a set of ordered pairs. b. ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ?
Inverse Relations and Functions Horizontal Line Test A function ? has an inverse function ? ? if and only if each horizontal line intersects the graph of the function in at most one point. If a function passes the horizontal line test, then it is said to be one-to-one, because no ?-value is matched with more than one ?-value and no ?-value is matched with more than one ? -value. .
Inverse Relations and Functions Horizontal Line Test If a function ? is one-to-one, it has an inverse function ? ? such that the domain of ? is equal to the range of ? ?, and the range of ? is equal to the domain of ? ?. .
Inverse Relations and Functions Sample Problem 2: Use a horizontal line test to determine whether of the graph of each function is a one-to-one function. a. ? ? = ?? ? y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Sample Problem 2: Use a horizontal line test to determine whether of the graph of each function is a one-to-one function. a. ? ? = ?? ? y 5 Because we can draw at least one horizontal line that intersects the graph more than once, ? ? = ?? ? is not a one-to-one function. Therefore the inverse of ? ? = ?? ? is a relation, but not a function. 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Sample Problem 2: Use a horizontal line test to determine whether of the graph of each function is a one-to-one function. b. ? ? = ??+ ? y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Sample Problem 2: Use a horizontal line test to determine whether of the graph of each function is a one-to-one function. b. ? ? = ??+ ? y 5 Because every horizontal line we draw intersects the graph only once, ? ? = ??+ ? is a one-to-one function. Therefore the inverse of ? ? = ??+ ? is a function. 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Step by Step Procedure to Find the Inverse of ? ? 1. Determine whether the function has an inverse by checking to see if it is one-to-one. 2. Replace ? ? with ?. 3. Interchange ? and ?. 4. Solve for ?.
Inverse Relations and Functions Step by Step Procedure to Find the Inverse of ? ? 5. Replace ? with ? ?? . 6. State any restrictions on the domain of ? ?? . Sometimes only part of the function you find algebraically may be the inverse function of ?. Therefore, be sure to analyze the domain of ? when finding ? ?.
Inverse Relations and Functions The graphs of the function and the inverse function are reflections across the line ? = ?.
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ? = ?? ? a.
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ? = ?? ? ? = ?? ? ? = ?? ? ? + ? = ?? ? + ? ? + ? = ?? ? + ? ? ? = ( , ) ? = ( , ) a. ? ?? =? + ? ? = ( , ) ? = ( , ) = ? ?
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ?? =? + ? ? ? = ?? ? a. ? y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ? = ? ?? b.
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? = ( , ) ? = ( , ) ? ? = ? ?? ? = ? ?? ? = ? ?? ? b. ?? = ? ?? ?? = ? ? ?? + ? = ? ? + ? ? = ( , ) ? = ( , ) ?? + ? = ? ? ?? = ?? + ?
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ?? = ?? + ? ? ? = ? ?? b. y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ? = ? ? + ? c.
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. c. ? ? = ? ? + ? ? = ? ? + ? ? = ?, ? = ?, ? = ? ? + ? ??= ?? ??= ? ? + ? ??= ?? + ? ?? ? ? ? = ?, ? = ?, ? ? + ? ? ?? =?? ? ? ? = ? ?
Inverse Relations and Functions Sample Problem 3: Find the inverse function, state any restrictions on its domain and then graph the function and its inverse. ? ?? =?? ? ? ? = ? ? + ? ? ? c. ? y 5 4 3 2 1 x -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5
Inverse Relations and Functions Compositions of Inverse Functions Two functions, ? and ?, are inverse functions if and only if: ? ?(?) = ?, for every ? in the domain of ?(?) and ? ?(?) = ?, for every ? in the domain of ? ? .
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? =? + ? ? ? = ?? ? a. ?
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? =? + ? ? ? = ?? ? a. ? ? ? ? = ? ? ? ? ? + ? ? ? ? ? = ? ? ? ? ? = ? + ? ? ? ? ? = ?
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? =? + ? ? ? = ?? ? a. ? =? ? + ? =?? ? + ? =?? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? = ??+ ?? + ? ? ? ? ? = ? + ? ? b.
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? = ??+ ?? + ? ? ? ? ? = ? + ? ? b. ? ? ? = ? ? + ? ? ??+ ?? + ? + ? ? ? ? ? = ??+ ?? + ? ? ? ? ? = ? + ?? ? ? ? ? = ? ? ? ? ? ? = ? + ? ? = ?
Inverse Relations and Functions Sample Problem 4: Show algebraically that ?and ? are inverse functions. ? ? = ??+ ?? + ? ? ? ? ? = ? + ? ? b. ?+ ? ? ? + ? ? ? ? = ? ? ?+ ? ( ? + ? ?) + ? ? ? ? = ? + ? ? ? ? ? = ? + ? ? ? + ? + ? + ? ? + ? ?? + ? ? ? ? = ?
