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Standards:

Standards:. MM2A5 – Students will explore inverses of functions. a. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range. c. Explore the graphs of functions and their inverses. Functions and Their Inverses.

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Standards:

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  1. Standards: • MM2A5 – Students will explore inverses of functions. • a. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range. • c. Explore the graphs of functions and their inverses.

  2. Functions and Their Inverses EQ: What are the characteristics of functions and their inverses?

  3. Relation: • Relation – a set of ordered pairs (or graph). • EX: { (0, 1) (-5, 3) ( ½ , 23) (.4, π) } or • Domain – x-values • Range – y-values

  4. Functions: • Function – a relation where every x-value is paired with exactly one y-value. (No x-values can repeat) • Vertical-line-test – If a vertical line intersects the relation's graph in more than one place, then the relation is NOT a function. No! Yes!

  5. Inverse of a Relation: • Inverse of a Relation – when a relation is taken and all the x-values and y-values have been switched. • The x and y coordinates have been switched. EX: relation: {(4, 10) (8, -2) (3, 5) (18, ½ )} inverse of the relation: {(10,4) (-2,8) (5,3) ( ½,18)}

  6. Graphically, the x and y values of a point are switched. If the function y = g(x) contains the points then its inverse, y = g-1(x), contains the points Y = x Where is there a line of reflection?

  7. Inverse Function: • Inverse Function – • The x and y-values have been switched and the resulting relation is a function. • Both the function and the inverse function are graphed and are symmetric over the y = x line. • Inverse Function Notation - f -1(x) (looks like f is raised to negative one, but is inverse notation)

  8. One-to-One: • When a relation is a function and its inverse is a function, then the original function is said to be one-to-one. • This means …every x-value is paired with exactly one y-value AND every y-value is paired with exactly one x-value. • Horizontal-line-test - If a horizontal line intersects the function’s graph in more than one place, then the function is NOT one-to-one. • (which means its inverse is not a function)

  9. Testing a Graph to see if it is One-to-One: 1. You must do the vertical line test to determine if the relation is a function. 2. Then, if it is a function, you can do the horizontal line test to determine if its inverse is a function. Yes! A function. Yes! One-to-One.

  10. Examples: Yes! A function. Yes! A function. Yes! One-to-One. No! Not One-to-One.

  11. Inverses to Linear Functions

  12. The inverse of a given function will “undo” what the original function did. 1. Look at the function F(x) and go through the order of operations as if you were replacing x with a value. 2. Now look at the inverse function F-1(x) and go through its order of operations. 1. Multiply by 2. 1. Subtract 5. 2. Add 5. 2. Divide by 2.

  13. Complete the table of values: Now Graph: 5 0 7 1 9 2 3 -1 1 -2

  14. Important facts about Inverses • If f is one-to-one, then f-1 exists. • The domain of f is the range of f-1, and the range of f is the domain of f-1. • If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f-1, so the graphs of f and f-1 are reflections of each other across the line y = x.

  15. Steps to Finding the Inverse to a Function: • 1. Replace f(x) with y. • 2. Switch the x and y. • 3. Solve the equation for y. • 4. Replace the y with f -1(x).

  16. Examples: • Find the inverse to the functions • 1. f(x) = 3x – 7 2. g(x) = ½ x + 10 y = 3x – 7 x = 3y – 7 + 7 + 7 x + 7 = 3y 3 3 x + 7 = y 3 f -1(x) = x + 7 3 y = ½ x + 10 x = ½ y + 10 - 10 - 10 x - 10 = ½ y 2(x – 10) = 2 ( ½ )y 2x – 20 = y g -1(x) = 2x - 20

  17. 5.1 Example of Finding f-1(x) Example Find the inverse, if it exists, of Solution Write f(x) = y. Interchange x and y. Solve for y. Replace y with f-1(x).

  18. Homework Sheet

  19. Inverses to Power Functions

  20. Inverses work for more than just linear functions. Let’s take a look at the square function: f(x) = x2 f -1(x) y x x f(x) 3 9 3 3 9 9 3 3 9 9 3 3 9 9 9 x2 3 3 9 9 3 3 9 9 3 3 9 9

  21. Power Functions: • When a function is written in the form f(x) = xn (where n > 1) it is considered a power function. • Examples of Power Functions: 1. f(x) = x2 2. g(x) = x3 3. h(x) = x4 ***How do we “undo” power functions?*** 1. 2. 3.

  22. Find the inverse to the Power functions: • 1. f(x) = x3 + 5 2. g(x) = (x – 1)3 - 2 y = x3 + 5 x = y3 + 5 x – 5 = y3 y = (x – 1)3 – 2 x = (y – 1)3 – 2 x + 2 = (y – 1)3

  23. How do I use composition of functions to verify that functions areinverses of each other? • If two functions, f(x) and g(x), are given and you need to verify that they are inverses… • You will have to show that the composition of • (f o g)(x) = x AND (g o f)(x) = x **This is because the two functions “undo” each other and what you put in for x in the original function will be undone by the inverse. Therefore, you will get x again.

  24. Example: • Verify that f(x) = 5x + 1 and g(x) = x – 1 are inverses 5

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