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Inverses of Functions

Inverses of Functions. Inverses of Functions. Objectives. Find the inverse of a relation or function Determine whether the inverse of a function is a function Demonstrate that two functions are inverses using composition of functions. A little review. y = (-1/4)x.

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Inverses of Functions

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  1. Inverses of Functions Inverses of Functions

  2. Objectives • Find the inverse of a relation or function • Determine whether the inverse of a function is a function • Demonstrate that two functions are inverses using composition of functions

  3. A little review y = (-1/4)x Solve each equation for y: 1. x = -4y 2. x = 2y + 3 3. x = (y + 3)/5 4. x= (-1/3)(y + 1) 5. Let f(x) = 2x – 4 and let g(x) = 0.5x + 2 Find (f ๐ g)(x) and (g ๐ f)(x). y = (1/2)x – (3/2) y = 5x - 3 y = -3x - 1 (f ๐ g)(x) = (g ๐ f)(x) = x

  4. What transformation maps flag ABCD to flag A’B’C’D’? What is line m called? The transformation is called a reflection and line m is the line of reflection.

  5. What line reflects BCD onto B’C’D’? How are the coordinates of the vertices of BCD related to the coordinates of B’C’D’? The line of reflection is the y-axis. The coordinates of the vertices of B’C’D’ have the same y-coordinates and x-coordinates with the opposite sign.

  6. What line reflects BCD onto B’’C’’D’’? How are the coordinates of the vertices of BCD related to the coordinates of B’’C’’D’’? The line of reflection is the x-axis. The coordinates of the vertices of B’C’D’ have the same x-coordinates and y-coordinates with the opposite sign.

  7. What line reflects BCD onto B’’C’’D’’? How are the coordinates of the vertices of BCD related to the coordinates of B’’C’’D’’? The line of reflection is the line y = x. The coordinates of the vertices of B’C’D’ have switched x-coordinates and y-coordinates with those in BCD

  8. The inverse of a relation consisting of the ordered pairs (x, y) is the set of all ordered pairs (y, x). • The domain of the inverse is the range of the original relation. • The range of the inverse is the domain of the original relation. • Therefore, to find the inverse of any relation or function,switch the domain and range. In the case of an equation, this means switch x and y.

  9. On a piece of graph paper plot the following points: (-2, 4), (3, -5), (0, 2), (-3, -4), and (2, 3) With a different colored pencil or pen plot points of the ordered pairs obtained by switching the x- and y-coordinates of each original pair. e.g. (-2, 4)  (4, -2) Now draw a line that is the line of symmetry between the two sets of points. What is the equation of that line? y = x

  10. Example 1 Find the inverse of the relation. State whether the relation is a function. State whether the inverse is a function. Relation: {(1,2), (2,4), (3,6), (4,8)} Inverse: {(2,1), (4,2), (6,3), (8,4)} Both the original relation and its inverse are functions.

  11. Example 1 Find the inverse of the relation. State whether the relation is a function. State whether the inverse is a function. Relation: {(1,5), (1,6), (3,6), (4,9)} Inverse: {(5,1), (6,1), (6,3), (9,4)} Neither the original relation not its inverse are functions. The original has 1 paired with two range values and the inverse has 6 paired with two range values.

  12. Example 2 Find an equation for the inverse of y = 3x - 2 To find the inverse of a relation, switch the domain and range. In this case that means switch x and y. y = 3x – 2 becomes x = 3y – 2 when x and y are switched. Now solve for y. x + 2 = 3y (Add 2) (x + 2)/3 = y (Divide by 3) y = (1/3)x + 2/3 (Write in y = mx + b form)

  13. Try this one Find the inverse of y = 4x - 5 Switch x and y and solve for y. x = 4y – 5 x + 5 = 4y (Add 5) (x + 5)/4 = y (Divide by 4) (1/4)x + 5/4 = y (Write in y = mx + b form)

  14. Inverse formulas The formula F = (9/5)C + 32 allows you to find the Fahrenheit temperature when the Celsius temperature is known. If we solve this formula for C, we have the inverse of the function. F = (9/5)C + 32 F – 32 = (9/5)C (Subtract 32) (5/9)(F – 32) = C (Multiply by the reciprocal) This formula allows us to find the Celsius temperature when the Fahrenheit temperature is known.

  15. Homework p. 122 (11 - 28, 29 - 34 find inverse only)

  16. Exploring the graphs of inverse functions. Graph y = 3x – 2, its inverse, and y = x. Use Zoom 4 for the viewing window. To do this put y1 = 3x – 2, y2 = x. To see the inverse press DRAW (2nd PRGM), 8 (DrawInv), Y1. Your graph should look like this.

  17. Exploring the graphs of inverse functions. Graph y = 3x + 2, its inverse, and y = x. Use Zoom 4 for the viewing window. To do this put y1 = 3x + 2, y2 = x. To see the inverse press DRAW (2nd PRGM), 8 (DrawInv), Y1. Your graph should look like this.

  18. Exploring the graphs of inverse functions. Graph y = -2x + 5, its inverse, and y = x. Use Zoom 3 for the viewing window. To do this put y1 = -2x + 5, y2 = x. To see the inverse press DRAW (2nd PRGM), 8 (DrawInv), Y1. Your graph should look like this.

  19. Exploring the graphs of inverse functions. Graph y = x2, its inverse, and y = x. Use Zoom 4 for the viewing window. To do this put y1 = -x2, y2 = x. To see the inverse press DRAW (2nd PRGM), 8 (DrawInv), Y1. Your graph should look like this.

  20. How does the graph of each function relate to its inverse function? Think about symmetry as you answer the question. Hopefully you noticed that the inverse of a relation or function is the reflection of the graph of the function across the line y = x. If the original graph was a function, is the inverse also a function? The graph of y = x2 is a function but its inverse is not a function. Notice it does not pass the vertical line test.

  21. If a function and its inverse are both functions, the inverse of f is denoted by f -1. When we graph the inverse of a function using the DrawInv command on the calculator, we cannot use the TRACE command on the inverse function. To do this we must graph the inverse in the Y= menu. Graph Y1 = 2x – 5 and Y2 = (1/2)x + (5/2). Use the TRACE command to find the x-intercept of Y1. How is it related to the y-intercept of Y2?

  22. To determine if the inverse of a function will also be a function, use the “Horizontal-Line Test” on the original function. For example, the function y = x2 would not pass the horizontal-line test so its inverse will not be a function. If you remember what the inverse function looked like, you will notice that the inverse function does not pass the vertical-line test.

  23. If a function has an inverse that is also a function, then the function is called a one-to-one function. Just as the graphs of f and f--1 are reflections of one another across the line y = x, the composition of a function and its inverse are related to the identityfunction. Theidentity function, I, is defined as I(x) = x. If f and g are functions and (f ๐ g)(x) = (g ๐ f)(x) = (I)x = x, then f and g are inverses of one another.

  24. Show that f(x) = 7x – 2 and g(x) = (1/7)x + 2/7 are inverses of each other. Because (f ๐ g)(x) = (g ๐ f)(x) = x, the two functions are inverses of each other. First find (f ๐ g)(x). (f ๐ g)(x) = f((1/7)x + 2/7) = 7 ((1/7)x + 2/7) – 2 = x + 2 – 2 = x Now find (g ๐ f)(x). (g ๐ f)(x) = g(7x – 2) = (1/7)(7x – 2) + (2/7) = x – (2/7) + (2/7) = x

  25. Show that f(x) = -5x + 7 and g(x) = -(1/5)x + 7/5 are inverses of each other. Because (f ๐ g)(x) = (g ๐ f)(x) = x, the two functions are inverses of each other. First find (f ๐ g)(x). (f ๐ g)(x) = f((-1/5)x + 7/5) = -5 ((-1/5)x + 7/5) +7 = x – 7 + 7 = x Now find (g ๐ f)(x). (g ๐ f)(x) = g(-5x + 7) = (-1/5)(-5x + 7) + (7/5) = x – (7/5) + (7/5) = x

  26. THE END THE END

  27. Homework: p. 122 (29 - 37, 41 - 46, 51 - 53)

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