1. INVERSE FUNCTIONS Section 3.3

2. Set X Set Y 1 2 3 4 5 2 10 8 6 4 Remember we talked about functions--- taking a set X and mapping into a Set Y An inverse function would reverse that process and map from SetY back into Set X 1 2 3 4 5 2 10 8 6 4

3. 1 2 3 4 5 2 8 6 4 If we map what we get out of the function back, we won’t always have a function going back!!!

4. Recall that to determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is a function This is NOT a function This is a function

5. If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs. So this does not have an inverse function! We can graph the inverse, but it is not a function. This does have an inverse function. Horizontal Line Test to see if the inverse is a function.

6. If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function. This is a one-to-one function This is NOT a one-to-one function

7. Find the inverse of a function equation : Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

8. Example 2: Given the function : y = 3x 2 + 2 find the inverse: Step 1: Switch x and y: x = 3y 2 + 2 Step 2: Solve for y:

9. Ex: Find an inverse of y = -3x+6. Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y

10. Consider the graph of the function The inverse function is

11. Consider the graph of the function The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y ! x x x x

12. x x x x is a reflection of in the line y = x What else do you notice about the graphs? x The function and its inverse must reflect on y = x

13. Graph f(x) and f -1 (x) on the same graph.

14. Double check to see if your right by seeing if it reflects over y = x

15. Graph: x f (x) -2 -8 -1 -1 0 0 1 1 2 8 Is this a function? Yes What will “undo” a cube? A cube root This means “inverse function” x f -1 (x) -8 -2 -1 -1 0 0 1 1 8 2 Let’s take the values we got out of the function and put them into the inverse function and plot them (2,8) (8,2) (-8,-2) (-2,-8)

16. Graph f(x) = 3x − 2 and using the same set of axes. Then compare the two graphs. Determine the domain and range of the function and its inverse.

18. Verify that the functions f and g are inverses of each other. If we graph (x - 2) 2 it is a parabola shifted right 2. Is this a one-to-one function? This would not be one-to-one but they restricted the domain and are only taking the function where x is greater than or equal to 2 so we will have a one-to-one function.

19. e.g.On the same axes, sketch the graph of and its inverse. x Solution:

20. Ex: f(x)=2x 2 -4 Determine whether f -1 (x) is a function, then find the inverse equation. f -1 (x) is not a function. y = 2x 2 -4 x = 2y 2 -4 x+4 = 2y 2 OR, if you fix the tent in the basement…

21. Ex: g(x)=2x 3 Inverse is a function! y=2x 3 x=2y 3 OR, if you fix the tent in the basement…

22. Exercise (d)Find and write down its domain and range. 1(a)Sketch the function where. (e)On the same axes sketch. (c)Suggest a suitable domain for so that the inverse function can be found. (b)Write down the range of.

23. (a) Solution: ( We’ll look at the other possibility in a minute. ) Rearrange: Swap: Let (d) Inverse: Domain: Range: (c) Restricted domain: (b) Range of :

24. Solution: (a) Rearrange: (d) Let As before (c) Suppose you chose for the domain We now need since (b) Range of :

25. Solution: (a) Swap: Range: (b) Domain: Range: (c) Suppose you chose for the domain Rearrange: (d) Let As before We now need since Choosing is easier!