Lesson video

Hello.

Mr. Robson here.

Welcome to maths.

Today, we're finding the inverse of a function.

Now this is a pretty awesome bit of maths.

So let's not hang around here.

Let's get stuck in.

Our learning outcome is that we'll be able to find the inverse of a function.

Some keywords you'll hear throughout the lesson.

Function, domain, and range.

A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

The domain of a function is the set of values that the mapping is performed on.

The range of a function is the set of values mapped to by the function and the stated domain.

Now this one might be new on you.

Inverse function.

An inverse function reverses the mapping of the original function.

In function notation, f inverse of x is the inverse function of f of x.

Look out for those words throughout today's lesson.

Two parts to our learning today.

We'll begin by finding the inverse of a function.

A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

For the function x times five, when mapping values of the domain x onto values of the range f of x, we use this notation.

We take f of x equals 5x.

And then when we're inputting into that function, when we want to input x equals negative one, we write it like this.

f of negative one, negative one goes into the function, i.

e.

, it's multiplied by five, and out comes negative five.

f of zero.

Five lots of zero.

The output is zero.

For f of one, the output is five.

For f of two, the output is 10.

And we can do that for any value of x.

When mapping multiple values and graphing, you can use a table of values.

Your table will look something like that.

The top row there, the x values, they're values of the domain.

The bottom row, f of x, represent values of the range, the outputs from the function.

Values of the domain are mapped to values in the range by multiplying by five for the function f of x equals 5x.

There are times when we'll need to map values of the range f of x back to the domain x.

For this same function, f of x equals 5x.

If f of x equals 20, what was the value of x that mapped to 20? I.

e.

, what did we need to put into the function to get 20 out? Well, if f of x equals 20, then 5x equals 20.

Oh, look, we have an equation to solve.

We'll divide both sides by five and find that x equals four.

Quick check you've got that and that you can find a few x values.

If f of x equals 75, what was the value of x that mapped to 75? And a second one for you, if f of x equals negative 40, what was the value of x that mapped to negative 40? In both cases, you're using the function f of x equals 5x.

Pause and see if you can find those two inputs.

Welcome back.

Hopefully, you started by setting up an equation.

If 5x equals 75, then x equals 75 divided by five.

x must equal 15.

What was the value of x that mapped to 75? It was 15.

For f of x equals negative 40, again, we can set up an equation.

5x equals negative 40.

Divide both sides by five.

x equals negative eight.

What was the value of x that mapped to negative 40? That value of x was negative eight.

In both examples, we had to divide by five to map back from the range to the domain.

In fact, to map any f of x value back, we can use x equals f of x divided by five.

This is known as the inverse function, as it reverses the mapping.

We write the inverse function using this notation.

f inverse of x equals x divided by five.

You'll more likely see that written as f inverse of x equals x over five.

A function maps values of the domain to values in the range.

The inverse function reverses the mapping.

It maps values of the range back to values in the domain.

In fact, the range of a function is the domain of its inverse function.

Quick check you've got that.

If f of x equals 12x, then can you select the correct inverse function for f of x? Three to choose from there.

Which one of those is going to reverse the mapping of f of x equals 12x? Pause and have a think.

Welcome back.

Hopefully you said C, f inverse of x equals x over 12.

Andeep and Sam are discussing function notation.

They're looking at the example we've just seen, f of x equals 12x, and f inverse of x equals x over 12.

Andeep says, "With exponents x to the power of negative one equals one over x.

Does that not mean that f inverse of x equals one over f of x?" What do you think about Andeep's question? Is he right? Sam knows.

In fact, Sam's got a cracking answer.

Sam says, "No.

This is function notation.

f inverse of x means f inverse of x.

We've seen an example like this before in sine inverse of x." Quick check you've got that.

For the function f of x equals x plus 10, can you select the correct inverse function from those three options? Pause and do this now.

Welcome back.

Hopefully, you went for option A.

And I wonder if you could pronounce that aloud? We pronounce it, f inverse of x equals x minus 10.

When you say an inverse function, be sure to say it like that.

We can check that an inverse function is correct by substituting a value into the original function.

If f of x equals 12x, then when we substitute in x equals four, we map four to 48.

Our function, 12x, maps four to 48.

We'll know we're correct if our inverse function maps 48 back to four.

So let's substitute 48 into our inverse function.

f inverse of 48 equals 48 over 12, which is four.

Well, hurrah! We know we're correct because our inverse function reverses the mapping.

Let's check you can do that.

f of x equals x plus 15.

What I'd like you to do is select the correct inverse function from below and substitute in any integer value to show that it works.

Pick any integer value, substitute it into f of x, and then take your output and substitute it into your inverse function and check.

Does it reverse the mapping? Pause and give this a go now.

Welcome back.

Hopefully, you picked B as the correct inverse function.

f inverse of x equals x minus 15.

In terms of substituting to show that it works, you can substitute in any integer value you like.

I've gone for x equals seven.

No reason.

You can use anything.

If I put x equals seven into the original function, f of seven becomes seven plus 15, that makes 22.

My inverse function is correct.

It should map 22 back to seven.

So I'll put 22 into my inverse function.

f inverse of 22 becomes 22 minus 15.

That equals seven.

Hurrah! We know we're right.

So long as your inverse function maps back to your original value, you are correct.

We'll frequently encounter functions that have multiple steps and we still need to be able to find their inverse.

If f of x equals 2x plus one, i.

e.

, x is being multiplied by two and then having one added, we can find the inverse function by making x the subject.

So to make x the subject, we need to add negative one to both sides and then we'll divide both sides by two.

We now see what needs to happen to f of x in order to map it back to the original x value.

The inverse function is to subtract one and then divide by two.

We'd write that like this.

The inverse function is x minus one divided by two.

f inverse of x is x minus one divided by two.

In maths, we're often presented with checking mechanisms to confirm whether we're right and we should absolutely use them.

I just showed you that the inverse of that function is x minus one over two.

We can check it works by substituting any integer value into the original function.

If our function f of x maps eight to 17, then our inverse function should map 17 back to eight.

So let's substitute 17 into the inverse function.

f inverse of 17 equals 17 minus one over two.

Wonderful.

That's eight.

We know it's worked because our inverse function has reversed the mapping from 17 back to eight.

Quick check you've got that.

I'd like you to make x the subject to find the inverse function and I'd like you to check your work by performing a substitution.

Pause and give this a go now.

Welcome back.

So finding the inverse function by making x the subject.

Hopefully, you multiplied both sides of the equation by two and then added one to both sides.

To map f of x back to our original x value, we must multiply by two, then add one.

Therefore, our inverse function is f inverse of x equals 2x plus one.

We can substitute now to show that this works.

I'm going to input nine into the original function and I can see that it maps to four.

What should happen now is four should map back to nine when I put it into our inverse function, and it does.

f inverse of four becomes two lots of four plus one.

That's nine.

Brilliant.

We know we're right.

Our inverse function reverses the mapping.

Practise time now.

For question one, I'd like you to match the functions to their inverse function.

There's four functions on the left-hand side.

You're going to match them to their inverse.

I've deliberately given you way too many choices.

You're going to have to decipher the correct four to select.

Good luck! Question two.

We're going to find some inverse functions.

What I'd like you to do is fill in the blanks to make x the subject and thus find the inverse function.

Two functions to work with there.

Pause and give this a go.

Question three, I'd like you to match the functions to their inverse function again.

The challenge this time is these functions are a little trickier.

You've got five functions on the left-hand side and I've given you six to choose from.

So one of them is going to be left out.

Pause now and see if you can pair those up.

Question four.

Part A.

I'd like you to show that f inverse of x equals seven bracket x minus four is the inverse function of f of x equals x over seven plus four.

Then for part B, I'd like you to show that f inverse of x equals 7x minus four is not the inverse function of f of x equals x over seven plus four.

Pause.

See if you can demonstrate those two things now.

Right, feedback time.

Matching functions to their inverse function.

f of x equals 9x, you should have matched to f inverse of x equals x over nine.

f of x equals x plus seven, you should have matched to f inverse of x equals x minus seven.

f of x equals x minus nine, we should have matched to f inverse of x equals x plus nine.

And finally, f of x equals x over seven should have matched to f inverse of x equals 7x.

Question two, I asked you to fill in the blanks to make x the subject and find the inverse functions.

That first blank in part A should show you adding 11 to both sides.

That'll be reflected in the next step.

Then both sides would have to be divided by four.

So what do we need to do to map back? We need to add 11 and then divide by four.

So our inverse function is f inverse of x equals x plus 11 over four.

For part B, that first missing step there was to subtract five from both sides.

That'll be reflected in the bracket in the next step.

What other step are we taking? Well, that would be to multiply both sides by three.

So if we need to take a function, subtract five, and multiply by three to map backwards, then f inverse of x equals three lots of bracket x minus five.

For question three, we were matching functions to their inverse function again.

These functions being a little more complicated.

You should have matched them up like so.

You might want to pause and check that your matching matches mine.

Question four, I asked you to show a couple of things.

Now, several ways we could have done this.

One way, a nice simple way, is to use substitution.

I can show when I substitute x equals 70 into the original function that it maps to 14.

If that is indeed the inverse function, then when I put 14 into the inverse function, it should map back to 70.

Wonderful.

The inverse function reverses the mapping.

We know it's right.

The opposite thing happens in part B.

If I substitute 70 into the original function, I still get an output of 14.

When I substitute 14 into that inverse function, 7x minus four, I get 94.

The function does not reverse the mapping.

Therefore, that's not the inverse.

Onto the second half of our lesson now where we're going to look at the graphs of functions and their inverse.

Let's consider the graphs of a function and its inverse.

Let's look at f of x equals 2x and its inverse function, f inverse of x equals x over two.

When we substitute one into the original function, we get an output of two, i.

e.

, one is mapped to two.

When we substitute two into the inverse, we get output of one, i.

e.

, our inverse function maps two to one.

Let's keep going with this and see what happens.

Let's input two into the function.

Our function maps two to four.

So let's input four into the inverse function.

Our inverse function maps four to two.

Can you see where this is going yet? Let's put three into the function.

Three maps to six.

Let's put six into the inverse.

Our inverse function maps six to three.

Quick check you're following this now.

I can put four into the original function and see that it maps four to eight.

What happens when I put eight into the inverse? Pause.

Have a think about this.

Tell the person next to you.

I'll see you in a moment.

Welcome back.

Funny enough, eight maps to four, and we get that coordinate pair there.

What do you notice about these two sets of coordinates? Pause.

Have a good look and a good think.

Welcome back.

I wonder what you noticed? When we look at the coordinates that were produced when we input into the function f of x, we had one two, two four, three mapped to six, four mapped to eight.

Because the inverse function reverses the mapping, we get a set of coordinates whereby the x-coordinate has become the y-coordinate and vice-versa.

If we draw the graph for y equals x, we can see that the set of points for the function and the set of points for the inverse function are reflections of each other in this line y equals x.

In fact, the graph of the inverse function is a reflection of the original function in the line y equals x.

There's our graph y equals f of x.

And there's our graph y equals f inverse of x.

What a beautiful reflection.

When we look at a table of values for the two functions, we notice something else.

That's a table of values for f of x.

The table of values for f inverse of x.

Have you spotted it? The domain of the original function became the range of the inverse function.

And the range of the original function became the domain of the inverse function.

Quick check you've got that.

What I'd like you to do is fill in the table of values for the inverse function and graph it.

I've given you all the values for the function f of x equals 4x minus one.

The inverse is f inverse of x equals x plus one over four.

Populate that table of values, and then what would it look like on that graph? Pause and do that now.

Welcome back.

Hopefully, your table of values looked like so, and then your graph looked like so.

There we have the graph y equals f of x and the graph y equals f inverse of x.

Now I'd like you to complete these two sentences.

There's a missing word in each.

What word are you going to put in those? Pause and think about that now.

Welcome back.

Your sentences should have read, the range of the function becomes the domain of the inverse function.

And the domain of the function becomes the range of the inverse function.

Finally, I'd like you to complete this sentence.

The inverse function is a what of the original function in the line what.

Pause.

Fill in those two blanks.

Welcome back.

Hopefully, your sentence read, the inverse function is a reflection of the original function in the line y equals x.

If I put the line y equals x on that graph, you can see that fact beautifully.

Andeep and Sam are discussing this graph.

Andeep says, "This symmetry in the line y equals x is amazing! It even works for quadratic functions like f of x equals x squared." But Sam says, "I can see a problem with your inverse function, Andeep!" Sam's right.

But can you see the problem? Pause now.

Have a good look.

What do you notice? Welcome back.

I wonder what you noticed? Andeep noticed something.

Andeep said, "Of course! It's a one-to-many mapping.

One input maps to many outputs.

That is not a valid function." Sam says, "Well spotted, Andeep!" I'd say, well done both of you.

If we reflect a many-to-one function such as f of x equals x squared in the line y equals x, we create a one-to-many relationship, which is not a valid function.

But we can get round this problem.

We can do that by restricting the domain of the original function.

For example, the function f of x equals x squared where x is greater than or equal to zero.

Look what happens when we graph that.

It's a one-to-one function.

The inverse function will be f inverse of x equals root x.

The domain of the function becomes the range of the inverse and vice-versa.

Hence, we get that table of values.

When we plot that inverse function, we've got a valid function.

By restricting the domain of the original function, we now have a valid inverse function which is still a reflection in the line y equals x.

Super quick check that you've got that.

True or false? All functions have an inverse function.

Is that true? Is it false? Once you've decided, could you justify your answer with one of those two statements at the bottom of the screen? Pause.

Have a think about this now.

Welcome back.

Hopefully, you said false, and justified that with the sentence, for some functions, we need to restrict the domain in order for their inverse function to be valid.

Practise time now.

Question one, I'd like you to draw the inverse function.

One assumption I'll give you is to assume the scales are the same on the x and y axes.

For part A, I've given you y equals f of x.

What would y equals f inverse of x look like? For part B, the same thing.

You've got y equals f of x.

What will the inverse function look like? Pause.

See if you can draw those now.

Question two, I'd like you to complete the table of values and draw the inverse function of f of x equals three lots of bracket x minus two.

Pause and do that now.

Question three, part A, I'd like you to draw the inverse function of f of x equals x squared plus three where x is greater than or equal to zero.

And for part B, I'd like you to write a sentence to explain what is wrong when we draw the inverse of f of x equals x squared plus three like so.

Pause and do those two things now.

Feedback time now.

Question one.

I asked you to draw the inverse functions.

We know the inverse function is going to be a reflection in the line y equals x.

So we need to reflect f of x in that line.

I'm going to pick a few coordinates.

Don't need to reflect all those coordinates.

A handful of coordinates correctly reflected will enable me to draw that line f inverse of x or y equals f inverse of x.

I'll do the same thing for part B.

There's y equals x.

I'll pick a few coordinate pairs to reflect and that'll enable me to draw that line, which is y equals f inverse of x.

You might want to pause.

Just check that your coordinate pairs and your lines match mine.

Question two, we were completing the table of values to draw the inverse function.

You should have remembered that the domain of the function becomes the range of the inverse and vice-versa.

So your table of values looks like so.

Once you've pulled up those coordinate pairs, you'll see the line y equals f inverse of x.

Question three, I asked you to draw the inverse function of f of x equals x squared plus three where x is greater than or equal to zero.

Again, it's a reflection in the line y equals x, so it should look like so.

There is y equals f inverse of x for that function.

For part B, I asked you for a sentence to explain what's wrong here.

You might have written, "Because a reflection in the line y equals x produces a one-to-many mapping.

This is not a valid function." We're at the end of the lesson now, sadly.

What have we learned? We've learned the inverse of a function can be found by rearranging to make x the subject.

We found the inverse function reverses the mapping of the original function.

And because of this feature, the graph of the inverse function is a reflection in the line y equals x.

We also learned that not all functions have an inverse function, as the inverse may be a one-to-many relationship, but it may be possible to restrict the domain in order to make the inverse function valid.

I hope you enjoyed today's lesson as much as I did.

I look forward to seeing you again soon for more mathematics.

Goodbye for now.