1. WARM UP 1. In the composite function m(d(x)), function d is called the ____________ function. 1. Give another symbol for m(d(x)). 2. If f(x) = 2x and g(x) = x + 3, find f(g(1)). 3. If f(x) = 2, find f(0). inside (m ° d)x g(x) = 1 + 3 = 4 f(4) = 2  4 = 8 2 = 1

2. INVERSE OF A FUNCTION

3. OBJECTIVES  Find an inverse relation and tell whether or not the inverse relation of the function is a function.  Identify the conditions necessary for a function to have an inverse function.  Graph a function and its inverse using parametric equations.

4. TERMS AND CONCEPTS  Inverse  Inverse function  Inverse function notation.  f and reflect across the line y = x  Invertible function  One-to-one function  Strictly increasing function  Strictly decreasing function  Parametric equations  Therefore symbol,  Q.E.D.

5. INVERSE FUNCTION  The photograph shows a highway crew painting a center stripe. From records of previous work the crew has done, it is possible to predict how much of the stripe the crew will have painted at any time during a normal 8-hour shift.  It may also be possible to tell how long it has been working by how much stripe has been painted. The input for the distance function is time, and the input for the time function is distance.  If a new relation is formed by interchanging the input and the output variables in a given relation, the two relations are called inverses of each other.  If both relations turn out to be functions, they are called inverse functions.

6. INVERSE OF A FUNCTION NUMERICALLY  Suppose that the distance d, in miles, a particular highway crew paints in an 8-hour shift is given numerically by this function of t, in hours, they have been on the job. t(h)d(mi) 10.2 20.6 31.0 41.4 51.8 62.2 72.6 83.0  Let d = f(t). You can see that f(1) = 0.2, f(2) = 0.6,…f(8) = 3. The input for function f is the number of hours, and the output is the number of miles.  As long as the crew does not stop painting during the 8-hour shift, the number of hours they have been painting is a function of the distance.  Let d = g(d). You can see that g(0.2) = 1, g(0.6) = 2,…f(3) = 8. The input for function g is the number of miles, and the output is the number of hours.  The input and output for functions f and g have been interchanged, and thus the two functions are inverses of each other.

7. SYMBOLS FOR THE INVERSE OF A FUNCTION  If function g is the inverse of function f, the symbol is often used for the name of function g.  In the highway stripe example, you can write (0.2) = 1, (0.6) = 2,…, (3) = 8. Note that (3) does not mean the reciprocal of f(3).  The -1 used with the name of a function means the function inverse, whereas the -1 used with a number, as in, means the multiplicative inverse of that number. and not 8

8. INVERSE OF A FUNCTION GRAPHICALLY  The figure shows a graph of the data for the highway stripe example. Note that the points seems to lie in a straight line.  Connecting the points is reasonable if you assume that the crew paints continuously. The line meets the t-axis at about t = 0.5, indicating that it takes the crew about half an hour at the beginning of the shift to redirect traffic and set up the equipment before they can start painting.

9. INVERSE OF A FUNCTION GRAPHICALLY  The second graph shows the inverse function t =. Note that every vertical feature on the graph of f is a horizontal feature on the graph of, and vice versa. For instance, the graph of meets the vertical axis at 0.5.  The third shows both graphs on the same set of axes. In this graph, x is used for the input variable and y for the output variable. Keep in mind that x for function f represents hours and x for function represents miles. The graphs are reflections of each other across the line whose equation is y = x.

10. INVERSE OF A FUNCTION ALGEBRAICALLY  In the highway stripe example, the linear function that fits the graph of function f in the third graph is y = 0.4(x – 0.5) or equivalently, y = 0.4x – 0.2 Slope = 0.4,x-intercept = 0.5. The linear function that fits the graph of f −1 is y = 2.5x + 0.5 Slope = 2.5, y-intercept = 0.5. If you know the equation of a function, you can transform it algebraically to find the equation of the inverse relation by first interchanging the variables. Function: y = 0.4x − 0.2 Inverse: x = 0.4y − 0.2

11. INVERSE OF A FUNCTION ALGEBRAICALLY  The equation of the inverse relation can be solved for y in terms of x. x = 0.4y − 0.2 y = 2.5x + 0.5 Solve for y in terms of x.  To distinguish between the function and its inverse, you can write f (x) = 0.4x − 0.2andf −1 (x) = 2.5x + 0.5  Bear in mind that x used as the input for function f is not the same as x used as the input for function f −1. One is time, and the other is distance.  An interesting thing happens if you take the composition of a function and its inverse. In the highway stripe example, f (4) = 1.4 andf −1 (1.4) = 4 f −1 (f (4)) = 4

12. INVERSE FUNCTIONS ALGEBRAICALLY  You get the original input, 4, back again. This result should not be surprising to you. The quantity f −1 (f (4)) means “How many hours does it take the crew to paint the distance it can paint in 4 hours?” There is a similar meaning for f (f −1 (x)). For instance, f (f −1 (1.4)) = f (4) = 1.4  In this case 1.4 is the original input of the inside function.