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1.5 Inverse Functions

1.5 Inverse Functions. Example 1 : An equation for determining the cost of a taxi ride is: C = 0.75 n + 2.50, where n is the number of kilometers and C is the cost. Complete the table of values. $10.00. $17.50. $25.00. $32.50. + 2.50. × 0.75. n. C. input number of kilometers.

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1.5 Inverse Functions

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  1. 1.5 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. Complete the table of values. $10.00 $17.50 $25.00 $32.50

  2. + 2.50 × 0.75 n C input number of kilometers output Cost – 2.50 ÷ 0.75 n C output number of kilometers input Cost 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 inverse function

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

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

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

  6. Example 2: Given the graph of y = f(x) below, sketch the graph of y = f–1(x). D: xÎÂ and R: y³ 0 f(2) = ? 8 Step 1: Sketch the graph of y = x. Step 2: Map the points using the line y = x as the axis of symmetry. f –1(2) = ? 0 Step 3: Join the points

  7. f(x) D: xÎÂ and R: y³ 0 f–1(x) D: x³ 0 and R: yÎÂ State the domain and range of f–1(x)

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

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

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

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

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

  13. b) Determine f (–6) c) Determine f –1(b + 1)

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

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

  16. Inverse of a Quadratic Function • A) Find the inverse of f(x)= x2-1 • B) Graph f(x) and its inverse. • C) Is the inverse of f(x) a function? • D) Determine the domain and range of f(x) and its inverse, i.e. f-1(x)

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