For our final unit, we're going to start by exploring Inverse Functions. Inverse functions are a way that we can undo what we do to an input. For example, if we doubled a number the inverse of that would be dividing it by 2. Before we get into the fancy notation, let's just look at a mapping between two sets. Here are three ordered pairs. The first set is a, b and c. And the second set is 2, 3, and 1. A maps to 2, b maps to 3, and c maps to 1. We could also show this using a mapping diagram. Here's the first set, containing a, b and c. And here's the second set, containing the numbers 1, 2 and 3. We said a maps to since every input has 1 distinct and unique output. In other words, every single output has one unique input that creates it. Now, let's consider a different function. This time this function will have the ordered pairs 2a, 3b and 1c. This function is the inverse function of this one. And we could express it with this mapping. Notice that here we had the inputs of 1, 2 and 3 here, here and here. Now these input correspond to different outputs. 1 corresponds to c, 2 corresponds to a, and 3 corresponds to b. Look carefully at these two functions and their mappings to figure out their relationship. What do you think is the relationship between a function and its inverse? Once you've taken some time to think about this, I want you to try and answer this next question.
If f of x is this function of ordered pairs, what would be the inverse of this function? When you write your inverse function here, write is as a set with curly braces on the ends. And then write your coordinate pairs seperated by commas.
The inverse of this function would be negative 1,5, 3, negative 2, and 2,6. Great thinking if you found these three coordinates. Now, what you want to notice is that the inputs and the outputs switch for the inverse function. In other words, notice how a, b, and c were always the input for the first function we looked at. Whereas in the inverse function, a, b and c were all outputs. Another way of thinking about it is that the x and y coordinate always switched. So, here for x I had a, and for y I had 2. In the inverse function the x value is 2 and the y value is a. This is also true for these other two points. The x value and the y value switched. The same was true for this last point. So, to find this inverse function we simply switch the x and the y values for each coordinate pair. That's how we wind up with this function. Keep in mind that inverse functions undo the original function. So, if I put in 5, I get an output of negative 1. So, to undo that, when I put in my negative 1 to my inverse, I want to get out the output of 5, the original input.
Let's look at the inverse of linear functions. We're going to find the inverse function of f of x equals 2 x plus 3. We've seen functions like these before. This is just a linear equation that crosses the y axis at 0, 3 since the y receptor's 3. The slope of this equation is 2, so we'll rise 2 and then we'll move to the right one unit. Now this is all the way back to unit three when we first learned about lines. But now what we're going to do is we're going to undo these operations. We know f of x is the same thing as y. This is just the output. So, now, the first thing we need to do to find the inverse function is to switch x and y. If we switch the variables x and y, then we get this new equation. X equal 2 y plus 3. Now we just need to solve for y to get our new inverse function. so you tell me, what would you get for y if you solved for it? In other words, what's the expression over here that involves x? As a hint, it should be a fraction.
Well to isolate y, we'll start by subtracting 3 from both sides. This will give us 2 y on the right side, and x minus 3 on the left side. Now we just divide both sides of our equation by 2 to get 1 y equal to x minus 3, all divided by
Now we really found the inverse function here. But notice that it's labeled as y. And that could get confusing since I already had a y up here. Which stood for my original function. Y was the output of this function. So in order to not confuse these two. Let's label this f inverse of x. So instead of having y equals the quantity x minus 3 divided by 2. We'll have f inverse of x equals x minus 3 divided by 2. Now, notice that this is actually a line too. If we divide 2 into each term, we'll have f inverse of x equals 1 half x minus 3 halves. negative 3 halves or negative 1 and a half is our y intercept. And slope hear would be positive 1 half. And I just want to caution you to be careful with this negative 1. This isn't an exponent like we normally think about it. This is just the notation for the inverse of a function. So if we take sum input x and we multiply it by 2 and add 3. To undo these operations we would have to subtract 3 first and then divide by 2. Now, that should make sense since we're just undoing the last operation we did to x and working backwards. So we need to undo add by 3, which is minus 3, and then we undo the multiplication of 2, which is divide by 2. Let's look closely at the graph of this function. And its inverse to see one more unique relationship. It's actually one we've already encountered. Here on this blue graph, I've graphed a line, y equals 2 x plus 3. We have a y intercept at 3. And a slope of up two units, right one unit to get our next point of 1, 5. Now, our inverse function was 1 half x minus 3 halves. Or we could have written the equation like this. X minus 3 all divided by 2. It's the same exact line. Notice that for this line, our y-intercept is at negative 3 halves, and the slope is positive 1, right 2. Positive 1, right 2. So again, this was our original function and then this on was the inverse. Now what I want to show you is this line y equals x. Here's the black line drawn in. Notice that the function and its inverse are reflected over the line y equals x. In other words if we fold this part over the black line, we would get this inverse function of our graph. And if we fold this part of the original function over our black line we get this part of the inverse function. If we look more closely at a point like 0.3, which is on our original function, we can find a corresponding point on the inverse function. This one's is what's so unique about inverse functions. The x and y-coordinates always change. So if we choose another point in the graph, like negative 1.5, 0 for our function, we can find it on our inverse function. That would be 0, negative negative 4, comma negative 5, we can find that on our inverse. On our inverse function it will be negative 5, negative 4. That point is right here.
So, this was one way that we could find the inverse function. First we switched the values of x and y, and then we solve for y and replace that with f inverse of x. Now there is a quicker way to find the inverse of a linear function. I could jet list out what I did to x. First I multiplied x by 2, and then I took that result and I added three. So, to undo these operations I would want to start with the results and do the inverse of this step. So, I start with the result, which is x, and I'm going to subtract 3 first. Then I'll take that result and divide it by 2. Notice that to find the inverse function, we first undo the last step that we performed for f of x. Then we work backwards and undo each of the steps before. We can see this for an input like x equals 1. We'll multiply the input by 2, which equals 2, and then we'll add 3 to that result, which equals positive 5. So, if this is a true inverse function. If I put in positive 5, I should get out the original input which was 1. So notice, we'll take 5, use that for the input of our inverse function. And first, we'll subtract 3, which equals 2. Next, we'll divide that result by 2, so 2 divided by 2, equals 1. So, notice how our inverse function undoes the operations that was performed on 1. Our original function with an input at 1 gave us an output of 5. And then our inverse function with an input of 5 gave us our original input which was 1.
So, let's see if you can find the inverse function of f of x equals 6x minus 5. Write that inverse here.
Here, the inverse function is x plus 5 divided by 6. Great thinking, if you found this inverse function. One way to find the inverse is to switch the x and y variables. Once we switch x and y, then we can solve for y. So, adding 5 to both sides, will get x plus 5 equals 6y. Then, we'll divide both sides by six, to get y equals x plus 5 divided by 6. And remember, our last step is to replace this y with f inverse of x, since this represents the inverse function. Now, the other way to reason through this answer is to think about what we did to x. We multiplied it by 6, and then subtracted 5. So we want to undo this subtraction of 5, and then the multiplication of 6. We always undo the last operation we performed. So first, we'll take x and add 5, and then we'll divide that quantity by 6. We'll do the opposite of multiplying by 6. This also will give us our inverse function.
So now that we know the inverse function of this function is x plus 5 divided by 6. What do you think is the inverse function when x equals 2? You can enter that answer here.
Here, the solution is 7 6ths, or 1 and 1 6th. Nice thinking if you found this fraction. Basically, we're just going to evaluate our inverse function at x equals 2. So we just plug in the value of 2, and for x. So 2 plus 5 equals 7. And I still have 6 for my denominator. So 7 6ths or 1 and 1 6th. Now, there was another way to solve this problem. If we remember that the x and y coordinates switch for the inverse function, then we can make the y coordinate be 2. In other words, we know the input for the inverse function would be 2. This means that it must be a y value for the original function. So, we could take our original function and let this output or this y value equal 2 and solve for x. Letting this f of x equal 2, we can now solve for x. We'll have 6x equal to 7 and then we'll divide both sides by positive 6 to get x equals 7 6ths. Since x is the input for the original function, we know this will be the output for our inverse function. Remember that the x and y coordinates will always switch for the function and it's inverse. Now, I know the second method is a lot longer for finding our answer, since all we really needed to do was plug in 2 into here. But it's worth conceptually noting, so that way we can understand that the inverse and its function always have the x and y coordinates switched.
Here the inverse is x + 4, all divided by +3. Great work if you found the inverse. We'll start by writing our function with the variables y for the output and x for the input. Now we want to switch x and y. This gives us a new equation that we can now solve for y to find our inverse function. First we'll add 4 to both sides to get x + 4 = 3y. Then we'll divide both sides by +3 to get y = (x+4)/3. We change y into f inverse of x to denote the inverse function. So here is our final answer.
Now, what do you think would be the inverse function for this one? When you think you've got it enter it here.
Now this wasn't just a linear function. This was a cubic function. And its inverse is the cubed root of x minus 1. Now that's great work if you found this solution. Again, we'll start by making f of x by y, our output. Now we're going to switch x and y. Now we have a function that we can solve for y in order to find the inverse. Now our goal is to get y alone, so first we'll subtract 1 from both sides of the equation, to get x minus 1 equals y cubed. Now we need to undo this power of 3. Well, I know cube-roots undo a third power. So taking the cube-root of both sides of the equation, I'll have one y equal to the cube-root of x minus one. Now, keep in mind this is our inverse function, so we take y and replace it with f-inverse of x. Here's our solution.