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

INVERSE OF A FUNCTION

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.

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

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.

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. 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. t(h) d(mi) 1 0.2 2 0.6 3 1.0 4 1.4 5 1.8 6 2.2 7 2.6 8 3.0 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.

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). and not 8 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.

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.

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.

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

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   and f −1(1.4) = 4    f −1(f (4)) = 4

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.

INVERTIBILITY & THE DOMAIN OF AN INVERSE RELATION In the highway stripe example, the length of stripe painted during the first half hour was zero because it took some time at the beginning of the shift for the crew to divert traffic and prepare equipment. The graph includes times at the beginning of the shift, along with its inverse relation. Note that the inverse relation has multiple values of y when x equals 0. Thus the inverse relation is not a function. You cannot answer the question, “How long has the crew been working when the distance painted is zero?” If the domain of function f is restricted to times no less than a half hour, the inverse relation is a function. In this case, function f is said to be invertible. If f is invertible, you are allowed to use the symbol for the inverse function.

INVERTIBILITY & THE DOMAIN OF AN INVERSE RELATION If the domain of f is 0.5 < x < 8, then there is exactly one distance for each time and one time for each distance. Function f is said to be a one-to-one function. Any one-to-one function is invertible. A function that is strictly increasing, such as the highway stripe function f, or strictly decreasing is a one-to-one function and thus is invertible. The highway stripe problem gives examples of operations with functions from the real world. Example 1 shows you how to operate with a function and its inverse in a strictly mathematical context.

EXAMPLE 1 Given f(x) = 0.5x + 2 Make a table of values for f(-2), f(-1), f(0), f(1) and f(2). From the numbers in the table, explain why you can’t find the value of x if f(x) = 2.5. How does this result tell the function f is not invertible? Plot the five points in part a on graph paper. Connect the points with a smooth curve. On the same axes, plot the five points for the inverse relation and connect them with another smooth curve. How does the graph of the inverse relation confirm that function f is not invertible? Find an equation for the inverse relation. Plot function f and its inverse on the same screen on your grapher. Show that the two graphs are reflections of each other across the line y = x.

SOLUTION Given y = 0.5x + 2 Make a table of values for f(-2), f(-1), f(0), f(1) and f(2). From the numbers in the table, explain why you can’t find the value of x if f(x) = 2.5. How does this result tell the function f is not invertible? If f(x) = 2.5, there are two different values of x, −1 and 1. You cannot uniquely determine the value of x. x f(x) -2 4 -1 2.5 2 1 Function f is not invertible because there will be two values of y for the same value of x if the variables are interchanged.

SOLUTION b. The graph shows function f and its inverse relation. The inverse relation is not a function because there are two values of y for each value of x > 2. the inverse relation fails the vertical line test. f Inverse of f

SOLUTIONS x = 0.5y + 2 y = 2x – 4 Given y = 0.5x + 2 c. Write the equation for the inverse relation by interchanging the variables and transforming the equation so that y is in terms of x. x = 0.5y + 2 Interchange x and y to get the inverse relation. y = 2x – 4 Solve for y Take the square root of each side Enter the two branches of the inverse relation as Enter y = x as The graph shows function f and its inverse relation. The graphs are reflections of each other across the line y = x

PARAMETRIC EQUATIONS There is a simple way to plot the graph of the inverse of a function with the help of parametric equations, equations in which both x and y are functions of some third variable, t. You will learn more about parametric equations in later chapters. Example 2 shows you how you can use parametric mode on your grapher to graph inverse relations.

EXAMPLE 2 Plot the graph of y = 0.5x + 2 for x in the domain -2 < x < 4 and its inverse using parametric equations. What do you observe about the domain and range of the function and its inverse? Solution: Put your grapher in parametric mode. Then, on the y menu, enter:  y1t = 0.5t2 + 2 Because x = t, this is equivalent to y = 0.5x2 + 2.               x2t = 0.5t2 + 2               y2t = t For the inverse, interchange the equations for x and y/ Use a window with a t-range of −2 ≤ t ≤ 4. Use a convenient t-step, such as 0.1. The range of the inverse relation is the same as the domain of the function, and vice versa. The range and the domain are interchanged.    

EXAMPLE 3 Let f(x) = 3x + 12 Find an equation for the inverse of f, and explain how that equation confirms that f is an invertible function. Demonstrate that f −1(f (x)) = x and f (f −1(x)) = x. Solution: Function y = 3x + 12 Inverse x = 3y + 12 y = 1/3x − 4 Because the equation for the inverse relation has the form y = mx + b, the inverse is a linear function. Because the inverse relation is a function, f is invertible, and you are allowed to write f −1(x) = 1/3x −4  b. f −1(f (x)) = f −1(3x + 12) Substitute 3x + 12 for f(x).              = 1/3(3x + 12) − 4 Substitute 3x + 12 as the input for function f −1  = x + 4 − 4 = x Show that f −1(f (x)) equals x. = 3 ( x − 4) + 12 Note: The three-dot mark stands for “therefore.” The letters Q.E.D. stand for the Latin words quod erat demonstrandum, meaning “which was to be demonstrated.”  = x − 12 + 12 = x.  f −1(f (x)) = x and f (f −1(x)) = x, Q.E.D.

DEFINITIONS & PROPERTIES The inverse of a relation in two variables is formed by interchanging the two variables. The graph of a relation and the graph of its inverse relation are reflections of each other across the line y = x. If the inverse of function f is also a function, then f is invertible. If f is invertible and y = f (x), then you can write the inverse of f asy = f −1(x). To plot the graph of the inverse of a function, either - Interchange the variables, solve for y, and plot the resulting equation(s), or Use parametric mode, as in Example 2. If f is invertible, then the compositions of f and f −1 are - f −1(f (x)) = x, provided x is in the domain of f and f (x) is in the domain of f −1  - f (f −1(x)) = x, provided x is in the domain of f −1 and f −1(x) is in the domain of f A one-to-one function is invertible. Strictly increasing or strictly decreasing functions are one-to-one functions.

CH. 1.5 ASSIGNMENT Textbook pg. 40 #1, 2, 3, 6, 8, 10, 22, & 27