1. Inverse Functions Lesson 2.4

2. 2 Back to the Magic Box What if we cram a number up the spout and out of the funnel pops the number that would have given us the result?? The function that does this is called the inverse function 1 -3 4 3 3 1 Use spreadsheet to evaluate inverse of a function

3. 3 Definition Recall that for function f, we write  The function f maps x 0 in domain  To y o in the range Consider another function f -1 which  Maps y o to x 0

4. 4 Example Suppose you are told 1 gallon of paint covers 250 ft 2  You might derive the function It is just as reasonable to consider how many gallons are needed for a certain area

5. 5 Example The mathematical relationship is the same  The input on one f(g) is the output on h(A) We would say the functions have an inverse relationship

6. 6 Inverse Function Notation For the inverse of function f, we use the notation f -1 Note that this is not the same as a negative exponent

7. 7 Finding Inverse Values from a Table Given the following table which defines the function f Determine  f(-2)  f -1 (2)  f -1 (-4)  f(-1) x-2012 f(x)6-4329

8. 8 Finding Inverse Values from a Graph Write some ordered pairs for the function defined by this graph Determine f -1 (0) f -1 (-2) xf(x) Are there multiple answers Is the inverse even a function? Are there multiple answers Is the inverse even a function?

9. 9 Finding the Inverse Formula Given the formula Find the inverse function f -1 (V) Strategy  Write in formula notation  Solve for the independent variable r = ?

10. 10 Domain and Range of An Inverse Function Note that the domain of the original function becomes the range of the inverse  Thus restrictions on the original domain affect the range of the inverse Also The range of the original may be restricted  This affects the domain of the inverse Consider the inverses of these functions As we saw on slide 7, some inverses might not even be functions

11. 11 Inverse Trig Functions Given y = sin (x) Table of values Graph xy = sin(x) -3.14160.0000 -2.6180-0.5000 -2.0944-0.8660 -1.5708 -1.0472-0.8660 -0.5236-0.5000 0.0000 0.52360.5000 1.04720.8660 1.57081.0000 2.09440.8660 2.61800.5000 3.14160.0000 3.6652-0.5000 4.1888-0.8660 4.7124 5.2360-0.8660 5.7596-0.5000 6.28320.0000 What if we reversed the ordered pairs … y for x ?

12. 12 Reversed Ordered Pairs Problem  This is not a function  Fails the vertical line test  There are multiple (x,y)'s where x =.5 Solution  Limit the range  xy 0.0000-3.1416 -0.5000-2.6180 -0.8660-2.0944 -1.5708 -0.8660-1.0472 -0.5000-0.5236 0.0000 0.50000.5236 0.86601.0472 1.00001.5708 0.86602.0944 0.50002.6180 0.00003.1416 -0.50003.6652 -0.86604.1888 4.7124 -0.86605.2360 -0.50005.7596 0.00006.2832

13. 13 The Inverse Trig Function We say Similarly for inverse cosine The range of cos -1 x is limited differently  Note pg 38 for domain, range of other functions

14. 14 Evaluating Inverse Functions Consider cos -1 (-0.5)  We are asking what angle has a cosine value of -0.5 Cosine negative in quadrants 2 and 3  But for cos -1 (x) we look only in 1 & 2 2 Calculator also capable of evaluating inverse trig functions

15. 15 Try It Out Consider these Note: newer calculators will have these functions – find in Catalog

16. 16 Assignment Lesson 1.4 Page 41 Exercises 3 – 49 odd