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Hi, I'm Mrs. Dennett.

And in this lesson, we're going to be finding inverse functions.

In this lesson, we're going to be looking at inverse functions.

And we write this as f to the minus one of x.

So, we're given a function of x, which is equal to five x, and we want to find the inverse.

Now, to do this, we're going to really need to be thinking about our order of operations and the hierarchy of operations.

So, remember brackets are our most important operation.

They always come first, followed by squares or square roots, cubes and cube roots, et cetera, so indices.

And then we've got multiplication and division, which are as important as each other.

And they are more important than addition and subtraction, which unfortunately, they're at the bottom of the pile, the most basic functions.

So, bearing that in mind, we're going to find the inverse function of five x.

So, we start by thinking about a function machine.

What is happening to x in our function of x? Well, x has been multiplied by five and we get five x.

Now, let's think about the inverse of that.

So, the inverse of multiplying by five will be to divide by five.

So, to get the inverse function, we imagine that we substitute x in and we would divide it by five, and we would get x over five.

Now, another way we can think about this is what's happening to x.

We divide it by five and we get x over five.

So, it's almost like making x the subject.

Let's have a look at another question.

So, here, we've got g of x, and we're going to find the inverse function of g of x or g to the minus one of x.

What's happening to x here? Well, x has been multiplied by three to give us three x.

We take away six, let's get three x minus six, and then we divide by four.

So, let's think about if we were going to put x into this function but in this order, we'd be multiplying it by four, adding six, and then dividing by three.

So, we'd get four x plus six divided by three.

And again, this is like making x the subject.

So, we could think about it like this.

We start with x, we times by four to get four x, we add six and we get four x plus six, and then we divide by three, which gives us four x, add six, all divided by three for our inverse function, g of x or g to the minus one of x.

So, now we're going to look at a function of h.

h of x is equal to five x squared plus eight, and this is a quadratic function.

And we want to find the inverse function of h of x, h to the minus one of x.

And in order to do that, we really need to be focusing on our order of operations again.

So, we think about what is happening to x in the function h of x.

Well, first of all, it's been squared, then it's been multiplied by five, and then we add eight.

And what we need to do is take these operations in reverse order, doing the inverse in order to find our function, h to the minus one of x.

So, the last thing that we did was add eight.

So, the first thing that we're going to do to x is we're going to take away eight.

Notice, we're doing the inverse.

So, we get x take away eight.

The next thing that we need to do is divide by five.

So.

we get x minus eight all divided by five.

And then finally, we're going to square root this fraction.

And we get our inverse function, h to the minus one of x is x minus eight, divided by five, all square rooted.

Now, we're going to look at what we can do in terms of using this inverse function.

So, we've got our inverse function, h to the minus one of x already in the first part of this question, and we want to substitute in the value of 33.

So, we take our value of 33, and we put it into the function and we get 33 take away eight, all divided by five and square rooted.

And that works out to be root five.

So, we can also substitute values into our inverse functions.

Here are some questions for you to try.

Pause the video to complete the task, and restart when you are finished.

Here are the answers.

I think the most complex answer here is question three because we have three operations to think about.

x has been multiplied by three, subtract five, and then divide by seven.

And if we want the inverse of this function, we reverse the order and apply the opposite operations.

So, our x is multiplied by seven, then we add five, and finally divide by three.

In part four, make sure that your square root sign incorporates all of x take away six, not just x.

So, the answer is square root, everything under that square root sign.

So, x minus six, all square rooted.

Here are some questions for you to try.

Pause the video to complete the task, and restart when you are finished.

Here are the answers.

In question five, x is divided by five, and then we subtract seven.

So, the inverse is to add seven, and then multiply by five.

And we want to multiply all of x plus seven by five.

And we use brackets to indicate this.

So, Gabrielle should have put brackets around x plus seven to show that all of that expression was being multiplied by five.

For question six, the inverse function, h to the minus one of x is equal to x take away six, all divided by eight.

And substituting 30 into this gives us three.

That's all for this lesson.

Remember, to take the exit quiz before you leave.

Thank you for watching.