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Slide 2 Inverse Functions By Dr. Marcia Tharp Slide 3 This module covers: An Intuitive Idea About Finding An Inverse Function. Making a 1 to 1 check if a function has an inverse Finding the inverse of a function. Graphing the inverse function. Slide 4 Finding the Inverse of a Function Take a look at the lists below. What do you notice? List 1 List 2 Slide 5 Thats right the x and y values places. So we let the List 1 represent a function f(x). List 2 will be the inverse of f(x) or f -1 (x) List 1=f(x) List 2= f -1 (x) If (x,y) is an ordered pair of f, then (y,x) is an ordered pair of the inverse f -1 (x) swap Slide 6 Looking for ordered pairs that are swapped or reversed is one way to intuitively find out if we have an inverse function. However we must be very careful when doing this. There is more... Slide 7 Some functions do not have inverses. Look at the y=x 2 or f(x)= x 2 and its inverse. f(x) =x 2 f -1 (x) Slide 8 In the second table we see that f -1 (x) has x=16 when y= -4 and also x =16 when y=4. Since there are two y values for the same x, f -1 (x) is not a function. So f(x)= x 2 does not have an inverse. f(x) =x 2 f -1 (x) Slide 9 Big Idea If this happens a function is one to one. For a function to have an inverse, looking at the first table each y-value of f must have exactly one x value Slide 10 Question Look at the lists below does this function have an inverse? F(x)F -1 (x) ?? Slide 11 Hint is F(x) one to one? Does each y value of F(X) have exactly one x value? F(x)F -1 (x) ?? Slide 12 Solution: F(X) is not one to one because 3 has two values associated with it 1 and 1. This means that the second list F -1 (x) is not a function. F(x) F -1 (x) ?? It is therefore not an inverse. Slide 13 Finding out if a function is 1 to 1 is easier to see in a graph. Lets look at the function g(x) below. g(x) Slide 14 Finding out if a function is 1 to 1 is easier to see in a graph. Lets look at the function g(x) below. g(x) and its graph. Slide 15 Remember when we used the vertical line test to see if we had a function. We checked each vertical line to see it touched the points only once. If this happened we were certain we had a function. G(X) is a function. Slide 16 Since we flip or reverse the points to get the inverse we are going to flip the line test to horizontal to see if g(x) is one to one. So if a horizontal line touches g(x) only once than g(x) is one to one. g(x) So g(x) has an inverse! Slide 17 Look at the list of ordered pairs and its graph. Does this function have an inverse? Hint: Check to see if the function is one to one. XY 0 01 1 23 32 4-2 Slide 18 Look at the list of ordered pairs and its graph. Does this function have an inverse? This function is one to one by the horizontal line test. Click to see it. XY 0 01 1 23 32 4-2 So it has an inverse! Slide 19 We know that we can the coordinates of a function to find the coordinates of the inverse. swap But what if the function is stated in rule form? How do we find the rule for the inverse? Slide 20 Remember that inverse rules undo each other. So if f(x)=x+5 the rule for adding 5 Then f -1 (x)=x-5 the rule for subtracting 5 is the inverse. But how do we find the inverse of r(x)=2x-1 ?? Likewise if g(x)= 3x the rule for multiplying by 3 then g -1 (x)= the rule for dividing by 3 is the inverse. Slide 21 Well here is a process to follow. Example: Find the inverse of r(x) = 2x-1 1)Replace the function symbol r(x) with y. y =2x 1 2) Solve for x. y +1 =2x 1 +1 Undo the 1 by using the opposite and adding +1 to get x by itself. y + 1 = 2x Simplify y + 1 = 2x 2 2 Undo the multiplication by 2 Using the opposite operation to divide by 2. y +1 = x 2 Slide 22 So y +1 = x 2 3) Reverse the order of the expressions around the equality. becomes x = y +1 2 4) Swap x and y just as we did with the points. y= x +1 2 5) Replace y with r -1 (x). r -1 (x). = x +1 2 This is the inverse of r(x) = 2x 1 Slide 23 You can check the inverse r -1 (x) mentally by comparing it to the original function r(x). See the next slide The operations on x in the inverse function should be the opposite of those in the original function. Slide 24 Original Function r(x) = 2x -1 Inverse Function r -1 (x) = x +1 2 Operations Addition of 1 Division by 2 Subtraction of 1 Multiplication by 2 We know we are on the right track because inverse uses opposites to undo the original function. We have addition of 1 in the inverse and its opposite subtraction of 1 in the original function And we see division of 2 in the inverse and its opposite multiplication of 2 in the original function. Slide 25 O.K. Its your turn! Find the inverse of f(x) = x-4 7 Slide 26 Solution: 1) Replace the function symbol f(x) with y in f(x) = x-4 7 y = x-4 7 2) Solve for x. 7y = (x-4) Multiply by 7 7y + 4 = xAdd 4. 3) Reverse the order of the expressions around the equality. x= 7y +4 Slide 27 4) Swap x and y just as we did with the points. y=7x+4 5) Replace y with f -1 (x). f -1 (x)=7x+4 This is the inverse of f(x) = x-4 7 Notice that the inverse uses opposite operations of the original function f(x). Slide 28 Now how do we make a graph of the inverse function when given a function g(x)? g(x) First lets swap x and y to get the inverse and see where the new points show up. Slide 29 Now lets graph the inverse points in red. g(x) Now draw a line segment to connect each point with its inverse. g -1 (x) Slide 30 Now draw in the line y=x g(x) What do you notice about the distance to the inverse? g -1 (x) Slide 31 To help draw in the line y=x g(x) What do you notice about the distance to y=x from the inverse? From the original point? g -1 (x) Slide 32 Thats right they are the same! g(x) As you look at the graph you will see that the diagonal distances are equal. g -1 (x) 2 2 3 3 2 2 1.5 So the points and the inverse points are reflections in the line y=x. Slide 33 This means we can graph the inverse function by using the line y=x. By finding the reflection of the point in the line y=x. For example lets graph the inverse of y=2x+1. 1) Draw the line y=x. 2) Locate points on y=2x + 1 3) Measure the distance to y=x. 4) Locate the reflected point that is the same distance away from y=x. 1 1.5 1 Connect the points to draw the inverse. inverse y=x Slide 34 Here are some links to interactive pages about inverses. Try these than go on to the practice problems listed in the file below this one in blackboard. Inverse Maker-Put in a function and it will give you the inverse. http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/inversefunction.def Inverse Point Grapher-Put in the coordinates of a point and this will show the point and its graph. http://www.livemath.com/lmstorage/files/10185353169168/inverse.html Good Luck!