Mathematics: Inverse Functions Note: You should know how to rearrange equations using algebra to make the most out of this lesson. Now, let’s say we have the function, f(x)= 3x+2 The inverse of the function is what we're looking for. I'll first solve for the inverse function, then discuss its significance. I won't go into detail about the inverse function's meaning just yet. So, to find the inverse function of a function, you simply have to follow the recipe listed below: 1) Change f(x) = y if required. 2) Change “x” into “y” and “y” into “x” 3) Use algebra to make “y” the subject again. Let’s try this out now with our original function f(x) = 3x+2 1) y=3x+2 2) x = 3y+2 3) This step is probably the trickiest as it requires the use of some basic algebra. We simply make “y” the subject and we can do it here by isolating it on the right side. To do this, simply: ● Take away 2 from each side ● Divide each side by three. As a result, you will get: y= (x-2)/3 Now, what’s so significant about the original function and its inverse? If you look closely, you will notice that the inverse function is simply the reflection of the original function along the function y=x. Let’s do a more difficult inverse problem. Find the inverse function of f(x)= x 2 +4x+4 1. y=x 2 +4x+4 2. Notice that we can use complete the square to turn x 2 +4x+4 into (x+2) 2 . If you’re lost, be sure to check out the previous tutoring about factorizing functions. 3. Now, y= (x+2) 2 . We can now turn “y” into “x” and “x” into “y” to get x=(y+2) 2 4. Time for some algebra. To make “y” the subject again, we can simply square root both sides to arrive at: √x = ±(y+2) Note: y+2 can be either positive or negative. Think about it like this. What is (-2) 2 and 2 2 . They’re both 4. So if you square root four, you can get either 2 or -2. The same concept applies here.
So: √x = ±(y+2) How can we make “y” the subject now? We first get rid of the brackets in the (y+2) and split it into two possible scenarios, with (y+2) both being positive and negative. Scenario 1: √x= +(y+2) Now deduct two from each side to get: y= √x – 2 Scenario 2: √x= -(y+2) √x= -y-2 Add 2 to each side to isolate “y” √x + 2 = -y-2 +2 -y = √x +2 Multiply each side by -1 (-1)(-y) = (-1)( √x+2) y= -√x-2 So the two answers are: y= √x -2 y= -√x -2 If we plot this graphically, we again notice that the inverse function is simply a reflection in the y-axis.