1. © The Visual Classroom 3.4 Inverse Functions Example 1: An equation for determining the cost of a taxi ride is: C = 0.75n + 2.50, where n is the number of kilometers and C is the cost. nC 10 20 30 40 $10.00 $17.50 $25.00 $32.50 Complete the table of values.

2. © The Visual Classroom The inverse function allows us to ask the question: if I have $50, how far can I travel? original function: C = 0.75n + 2.50 × 0.75 + 2.50 n C input number of kilometers output Cost inverse function ÷ 0.75 – 2.50 n C output number of kilometers input Cost

3. © The Visual Classroom A B f 6 23 2 4 7 15 Domain: {2, 4, 6} Range: {7, 15, 23} We know that a function maps elements of a domain onto elements of a range. A B 6 23 2 4 7 15 f –1 The inverse function maps the elements of the range back onto the elements of the domain. Domain: {7, 15, 23} Range: {2, 4, 6}

4. © The Visual Classroom Assume we have a function which consists of a set of ordered pairs. f(x) = {(1, 5), (2, 9), (3, 13), (4, 17)} Domain: {1, 2, 3, 4}Range: {5, 9, 13, 17} f –1 (x) = {(5, 1), (9, 2), (13, 3), (17, 4)} Range: {1, 2, 3, 4} Domain: {5, 9, 13, 17} f –1 (x) means the inverse function of f(x).

5. © The Visual Classroom Suppose we have the following relation f (x) consisting of the following points. Determine the graph of f –1 (x). We have symmetry about the line y = x. f(– 4) = f(0) = ? ? 2 5 f –1 (1) = ? f –1 (0) = ? –2 –6

6. © The Visual Classroom Example 2: Given the graph of y = f (x) below, sketch the graph of y = f –1 (x). Step 1: Sketch the graph of y = x. Step 2: Map the points using the line y = x as the axis of symmetry. Step 3: Join the points D: x  and R: y  0 f (2) = ?8 f –1 (2) = ?0

7. © The Visual Classroom f –1 (x) D: x  0 and R: y  f(x)f(x) D: x  and R: y  0 State the domain and range of f –1 (x)

8. © The Visual Classroom f(x) = 3x + 4Example 3: Given the equation: Determine the equation of f – 1 (x). y = 3x + 4 f f –1 1. Replace y by x and x by y. 2. Isolate y. x – 4 = 3y x = 3y + 4

9. © The Visual Classroom Compare f and f –1 and the order in which operations are carried out. 1. Multiply by 3 2. Add 4 1. Subtract 4 2. Divide by 3 y = 3x + 4 f f –1 You will notice the order and the operations are inverted.

10. © The Visual Classroom When you go to bed at night 1- you untie your laces 2- you take off your shoes 2- you put on your shoes When you get up in the morning 1- you put on your socks Notice the inverse operation 3- you take off your socks 3- you tie up your laces Reverse order … reverse operation

11. © The Visual Classroom 1. Multiply by 5 2. Subtract 2 1. Add 2 2. Divide by 5 f(x) = 5x – 2 f –1 Example 4: Determine the inverse of f (x) = 5x – 2 x + 2 Ex: f(4) = 5(4) – 2 = 20 – 2 = 18 f –1 (4) =

12. © The Visual Classroom Example 5: Given: a) Determine f –1 (x) replace x by y and y by x. × 2 isolate y.

13. © The Visual Classroom b) Determine f (–6)c) Determine f –1 (b + 1)

14. © The Visual Classroom Example 6: the relation f is defined by 2x – 3y = 6. Graph f xf(x)f(x) Using the intercept method. 0 – 2 0 3 Graph f –1 f -1 (x)x 0 – 2 0 3 f f –1

15. © The Visual Classroom The relation f is defined by 2x – 3y = 6. f f –1 Determine: f(–3) = – 4 f –1 (–3) = – 1.5 f –1 (x): 2y – 3x = 6 2y = 3x + 6