In this video, we’re gonna look at some functions and their graphs. And we’re gonna talk about the concept of a function and its inverse. And we’ll also look at the algebraic steps to find the inverse of a function and try out this technique on a few typical questions.
First, let’s look at the function 𝑓 of 𝑥 is equal to a half 𝑥 plus three. And to work with this function, first of all we input an 𝑥-value. Then the function tells us to half that number and add three to the result. And out pops the answer. And if we were to plot the graph of 𝑦 equals 𝑓 of 𝑥, it would look like this. Where the line represents the mapping of all possible input values onto their corresponding output values. And we can use this graph to work out the 𝑦-values from a given 𝑥-value as well. So if we were given an input value, say four, we could trace it up to the line and then across to the 𝑦-axis to get the corresponding 𝑦-coordinate or function output value. So if we’re given a value of 𝑥, we can use the function to find the corresponding 𝑦-value, either on the graph or using the equation 𝑦 equals a half 𝑥 plus three.
Now, it’s important to remember that in this particular function, any value, any real value of 𝑥 is valid for input. The domain or the set of numbers which you can input into this function is all of the real numbers. And because that line goes off infinitely in this direction and in this direction. The range or set of possible outputs is also the complete set of real numbers on the 𝑦-axis. Now we’ll come back to those two things, the range and domain, later on in this video.
Going back to the function we were looking at, what if we were given a 𝑦-value and we wanted to find the corresponding 𝑥- or input value? Well, we could read backwards on the graph. So we can look at our value on the 𝑦-axis. We can go across to the line and then down and simply read off the corresponding 𝑥-coordinate. But it would be great to have a nice equation that we could use without having to draw the graph. Well, of course, we can use our algebra skills to rearrange the original equation and make 𝑥 the subject. So if 𝑦 is equal to a half 𝑥 plus three, I could subtract three from both sides to get 𝑦 minus three equals a half 𝑥 plus three minus three. Well, those two things cancel out. So that’s 𝑦 minus three is equal to half 𝑥. Then I can multiply both sides by two to give me two 𝑦 minus six is equal to 𝑥 or 𝑥 is equal to two 𝑦 minus six.
And we can simply substitute values in for 𝑦. Do the calculation. And find the corresponding value of 𝑥. So this process and these ideas are the basic foundation of inverse functions. Now, we saw how the graph mapped all of the input values from the domain into the corresponding output values in the range. Well, the inverse function maps all of the original range values back to the original domain values. Now, if we could flip over the 𝑥- and 𝑦-axes, we could define a new function which maps what was the range onto what was the domain. This is how we find the inverse function. It’s kind of the reverse of the original function.
So here’s the graph of the original function. Now let’s add the line 𝑦 equals 𝑥. Then we can reflect everything in the line 𝑦 equals 𝑥. Which effectively swaps over the axes and gives us a new function 𝑦 equals the inverse of the original function. So the function and its inverse are mirror images of each other in the line 𝑦 equals 𝑥. So notice how two maps to four on the original function and four maps back to two on the inverse function. That’s what inverse functions are all about.
And briefly going back to the domain and range. What was the domain for the original function has now been swapped over onto the 𝑦-axis for the inverse function. So the domain from the original function becomes the range of the inverse function. And what was the range for the original function becomes the domain for the inverse function. Right, let’s go through a few examples and just talk this through. So first, let’s just finish off this example that we’ve been looking at.
Find the inverse function for 𝑓 of 𝑥 equals a half 𝑥 plus three.
Now, probably, the easiest way to approach these questions is to swap 𝑥 and 𝑦 in the equation. And then rearrange to make the new 𝑦 the subject. So swapping over the 𝑥’s and 𝑦’s, we’ve now got 𝑥 is equal to a half 𝑦 plus three. And just like we did before, we can now unpick that. So subtracting three from both sides, we’ve got x minus three is equal to a half y. And then doubling both sides, we’ve got two 𝑥 minus six is equal to y. And that gives us our inverse function. This is the function that maps all of the old outputs back to their original inputs. It’s the equation of this line here, our inverse function line.
Now, find the inverse function given that 𝑓 of 𝑥 is equal to three 𝑥 minus two.
So write 𝑦 equals three 𝑥 minus two, swap 𝑥 and 𝑦 in the equation, and then rearrange to make 𝑦 the subject. And that gives us our inverse function. So both the examples we’ve looked at so far have had domains and ranges which have included all the real numbers. So they’ve been relatively easy questions to do. But when you see slightly more complicated questions, sometimes you have to think carefully about the domain and the range.
Find the inverse function given that 𝑓 of 𝑥 is equal to three plus the cube root of 𝑥.
Once you start seeing things like cube root and square root, then you have to be aware. But, in fact, with cube root, again, a domain and a range contain all of the real numbers. So we don’t have to worry. So we write out our equation, 𝑦 equals three plus the cube root of 𝑥, swap the 𝑥- and 𝑦-variables, and then rearrange to make 𝑦 the subject. And, again, that gives us our inverse function, the function which maps the original 𝑦-values back onto their original 𝑥-values.
Now, we’ve got to determine the inverse of the function 𝑓 of 𝑥 is equal to the square root of 𝑥 plus three.
Well, there’re a couple of things we need to think about here. The square root function is defined as only being the positive square roots. And given that we can’t find the square root of negative numbers, there are no real solutions. The 𝑥-values here must be greater than or equal to zero. So the domain for this function must be 𝑥 is greater than or equal to zero. Now, if 𝑓 of 𝑥 were just equal to the square root of 𝑥, the graph would look something like that. But we’re adding three to all of those values. So that moves the curve up to look something like this. So our domain was that 𝑥 had to be greater than or equal to zero. But that has the knock-on implication that the 𝑦-values are gonna end up being greater than or equal to three. So we’ll just need to bear that in mind when we’re thinking about our answer.
So, again, we’re gonna start off by just writing out the equation of the function. So 𝑦 is equal to the square root of 𝑥 plus three. Then we’re gonna swap the 𝑥- and 𝑦-variables over. And then we’re gonna rearrange to make 𝑦 the subject. So we can subtract three from both sides and then square both sides. So that gives us our inverse function of 𝑦 equals 𝑥 minus three all squared. Or 𝑓 inverse of 𝑥 is equal to 𝑥 minus three all squared. But, remember, that process of rearranging and swapping the 𝑥- and 𝑦-variables was actually all about reflecting in the line 𝑦 equals 𝑥. So this is the curve we were looking for. But 𝑦 equals 𝑥 minus three all squared actually carries on up here, like this. So we don’t really want this bit.
But remember, the range of the original function maps onto the domain of the inverse function. So if we set our new domain to be 𝑥 is greater than or equal to three, then we’re only looking at the bit of that curve that we’re interested in. Now, remember, we swapped 𝑥 and 𝑦 when we were doing the calculation. So we swap 𝑥 and 𝑦 in the domain and range as well. This will be our answer. The inverse function of 𝑥 is 𝑥 minus three all squared. And that only works for 𝑥 is greater than or equal to three. Okay then, one last example.
Determine the inverse of the function 𝑓 of 𝑥 is equal to 𝑥 minus two all squared minus three where 𝑥 is greater than or equal to two.
Now, before we attempt this question, let’s try to sketch the function that we’re starting off with. So we’re gonna start off with it looks like it’s some sort of variation on 𝑥 squared. So if we start off with the 𝑥 squared function, then this bit in these parentheses here is a translation of 𝑦 equals 𝑥 squared by positive two in the 𝑥-direction. And then we’re taking away three from all of those results, the 𝑦-coordinates. So that’s a translation of negative three in the 𝑦-direction. So doing this one first, we’ve got 𝑦 equals 𝑥 minus two all squared would look like that 𝑦 equals 𝑥 squared shifted right two. And then shifting it down three is gonna look something like, oops, something like that. So that minimum point there would move down to negative three. But it would still have an 𝑥-coordinate of two.
But the question said that the domain was only four 𝑥 is greater than or equal to two. So that’s this part of the curve here going up. So we can rub out everything else. So the resulting function is just gonna look like that. Now let’s think about our domain and range. Well, the question told us that the 𝑥-values we could use for inputs were greater than or equal to two, so our domain is greater than or equal to two. And this is only generating answers which are greater than or equal to negative three. So the range can only contain answers of 𝑦 is greater than or equal to negative three. Now, having thought about that, we can go on and try and solve the question.
We start off by writing 𝑦 is equal to 𝑥 minus two squared minus three. Then we swap over the 𝑥- and 𝑦-variables and rearrange to make 𝑦 the subject. So, first, we can add three to both sides. Then we can take the square roots of both sides. And then, finally, add two to both sides. And having swapped over our 𝑥- and 𝑦-variables, we now need to swap over the domain and the range. So now the domain is not 𝑥 is greater than or equal to two. But it’s greater than or equal to negative three. What was the range of the 𝑦-values on the input function.
Now, if we think about the reflection in 𝑦 equals 𝑥, then for our inverse function, obviously the 𝑥-values that we can input are greater than or equal to negative three. And if we were asked for the range, the output values we could get would be greater than or equal to two. So our final answer then is that the inverse function 𝑓 inverse of 𝑥 is equal to the square root of 𝑥 plus three or plus two, for 𝑥 is greater than or equal to negative three.