3. EQ: What are the characteristics of functions and their inverses?

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

5.  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 a NOT a function.

6.  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 )} EX: relation: {(4, 10) (8, -2) (3, 5) (18, ½ )} inverse of the relation: {(10,4) (8,-2) (5,3) ( ½,18)}

7. 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 x01234y124816 x124816y01234 Where is there a line of reflection?

9.  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)

10.  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)

11. 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.

13.  All “sloping” linear functions are one-to-one.  All cubing functions are one-to-one.  Squaring functions (parabolas) are NOT one- to-one.  Absolute Value functions are NOT one-to-one (for same reason as parabola).  Square Root functions can be one-to-one, but the inverse has to have a restricted domain...

15. Imagine functions are like the dye you use to color eggs. The white egg is put in the function blue dye and the result is a blue egg.

16. The Inverse Function “undoes” what the function does. The Inverse Function of the BLUE dye is bleach. The Bleach will “undye” the blue egg and make it white.

18. F(x) = 2x + 5 F -1 (x) = x – 5 2 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. 2. Add 5. 1. Subtract 5. 2. Divide by 2.

19. F(x) = 2x + 5 F -1 (x) = x – 5 2 XYXY 05 17 29 3 -21 5 7 9 3 1 0 2 Now Graph: 1-2

20.  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).

21.  Find the inverse to the functions  1. f(x) = 3x – 72. 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

22. 5 5 5 5 5 5 25 25 25 25 25 25 25 25 25 25 5 5 5 5 5 5 5 5 5 Let’s look at another example for f(x) = x 2 x f(x) y Inverse x2x2 x

23. Function:Inverse Function: f(x) = x 2 f -1 (x) = √x XY XY 3993

24. Function:Inverse Function: f(x) = x 2 f -1 (x) = √x XY XY 3 9 9 3 525 5 What do you notice about the ordered pairs?

26. Inverses work for more than just linear functions. Let’s take a look at the square function: f(x) = x 2 3 x f(x) 3 3 3 3 3 9 9 9 9 9 9 9 y f -1 (x) f -1 (x) 9 9 9 9 9 99 3 3 3 3 3 3 3 x2x2 x

27. WWhen a function is written in the form f(x) = x n (where n > 1) it is considered a power function. EExamples of Power Functions: 1. f(x) = x 2 2. g(x) = x 3 3. h(x) = x 4 ***How do we “undo” power functions?*** 1. 2.3.

28.  1. f(x) = x 3 + 5 2. g(x) = (x – 1) 3 - 2 y = x 3 + 5 x = y 3 + 5 x – 5 = y 3 y = (x – 1) 3 – 2 x = (y – 1) 3 – 2 x + 2 = (y – 1) 3

29. EQ: How do I find the composition of functions?

30.  When we substitute one function into another one, it’s a composition of functions.  It essentially lets us perform two steps in one.  Notation:  Given f(x) and g(x)  (f o g)(x) can be written f(g(x)) …which means to substitute the function g(x) into f(x) everywhere you see an x.  (g o f)(x) – can be written g(f(x))… which means to substitute the function f(x) into g(x).

31.  Review:  Given f(x) = 3x + 4 and g(x) = ½ x – 5  1. Find f(-3) =  2. Find g(4) =  3. Find f(g(4)) =  4. Find g(f(10)) = f(-3) = 3(-3) + 4 = -5 g(4) = ½ (4) - 5 = -3 f(g(4)) = -3 f(-3)= -5 f(10) = 3(10) + 4 = 34 g(34) = ½ (34) – 5 = 12

32.  Given f(x) = 3x + 4 and g(x) = ½ x – 5, find…  1. (f o g)(x) =  f(g(x)) =  2. (g o f)(x) =  g(f(x)) = f( ½x – 5) = 3( ½x – 5) + 4 = 1.5x - 15 + 4 = 1.5x - 11 g(3x + 4) = ½ ( 3x + 4) - 5 = 1.5x + 2 - 5 = 1.5x - 3

33.  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.

34.  Verify that f(x) = 5x + 1 and g(x) = x – 1 are inverses 5