Inverse Functions By Dr. Marcia Tharp 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. Finding the Inverse of a Function Take a look at the lists below. What do you notice? List 1 List 2 That’s right the x and y values swap 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) If (x,y) is an ordered pair of f, then (y,x) is an ordered pair of the inverse f-1(x) List 1=f(x) List 2= f-1(x) 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 . . . Some functions do not have inverses. Look at the y=x2 or f(x)= x2 and its inverse. f(x) =x2 f-1 (x) 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)= x2 does not have an inverse. f(x) =x2 f-1 (x) Big Idea For a function to have an inverse, looking at the first table each y-value of f must have exactly one x value If this happens a function is one to one. Question Look at the lists below does this function have an inverse? F(x) F-1(x) ?? Hint is F(x) one to one? Does each y value of F(X) have exactly one x value? F(x) F-1(x) ?? 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. It is therefore not an inverse. F(x) F-1(x) ?? 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) 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. 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. 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. So g(x) has an inverse! g(x) Look at the list of ordered pairs and it’s graph. Does this function have an inverse? Hint: Check to see if the function is one to one. X Y 4.0 -1 0 3.0 0 1 2.0 1 -1 1.0 2 3 3 2 -1.0 4 -2 -2.0 -2.0-1.0 1.0 2.0 3.0 4.0 5.0 Look at the list of ordered pairs and it’s graph. Does this function have an inverse? This function is one to one by the horizontal line test. Click to see it. X Y 4.0 -1 0 3.0 0 1 2.0 1 -1 1.0 2 3 3 2 -1.0 4 -2 -2.0 -2.0-1.0 So it has an inverse! 1.0 2.0 3.0 4.0 5.0 We know that we can swapthe coordinates of a function to find the coordinates of the inverse. But what if the function is stated in rule form? How do we find the rule for the inverse? 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. Likewise if g(x)= 3x the rule for multiplying by 3 then g-1(x)= x the rule for dividing by 3 is the inverse. 3 But how do we find the inverse of r(x)=2x-1 ?? 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 y +1 = x 2 Undo the multiplication by 2 Using the opposite operation to divide by 2. 3) Reverse the order of the expressions around the equality. So becomes x = y +1 y +1 = x 2 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 You can check the inverse r-1(x) mentally by comparing it to the original function r(x). The operations on x in the inverse function should be the opposite of those in the original function. See the next slide Operations Inverse Function r-1(x) = x +1 2 Original Function r(x) = 2x -1 Addition of 1 Division by 2 Subtraction of 1 Multiplication by 2 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. We know we are on the right track because inverse uses opposites to undo the original function. O.K. It’s your turn! Find the inverse of f(x) = x-4 7 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) 7y + 4 = x Multiply by 7 Add 4. 3) Reverse the order of the expressions around the equality. x= 7y +4 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). Now how do we make a graph of the inverse function when given a function g(x)? First lets swap x and y to get the inverse and see where the new points show up. g(x) Now let’s graph the inverse points in red. Now draw a line segment to connect each point with its inverse. g(x) g-1(x) Now draw in the line y=x What do you notice about the distance to the inverse? g(x) g-1(x) To help draw in the line y=x What do you notice about the distance to y=x from the inverse? From the original point? g(x) g-1(x) That’s right they are the same! So the points and the inverse points are reflections in the line y=x. As you look at the graph you will see that the diagonal distances are equal. g(x) 1.5 2 g-1(x) 1.5 3 2 3 2 2 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. y=x 1) Draw the line y=x. 1.5 2) Locate points on 1 y=2x + 1 1.5 1 3) Measure the distance to y=x. 4) Locate the reflected point that is the same distance away from y=x. inverse Connect the points to draw the inverse. 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 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 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 Good Luck!