Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Nice of you to join me.

Today we're solving equations involving composite functions, which sounds awesome, let's take a look.

Our learning outcome is I'll be able to solve equations involving composite functions.

Some key words we'll hear throughout today's learning, function and equation.

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

An equation is used to show two expressions that are equal to each other.

Two parts to our learning today.

Let's begin by solving with composite functions.

If f of x equals x plus 8, and g of x equals 4x, we can solve f of x equals 20 and g of x equals 20.

If f of x equals 20, then x plus 8 equals 20, so x equals 12.

If g of x equals 20, then 4x equals 20, i.

e.

, x equals 5.

We can also solve equations formed from composite functions.

For example, f of g of x equals 20.

Well, we know an expression for f of g of x would be 4x plus 8.

That's a composite function, f of g of x.

So to solve f of g of x equals 20, we need to solve this.

4x plus 8 equals 20.

4x must equal 12, x must equal 3.

We can check that that's correct.

We can evaluate f of g of 3, and it successfully maps to 20.

When we solved f of g of x equals 20, we were looking for the input that maps to 20.

By checking that fg of 3 evaluated to 20, we've proven that that is indeed the input that maps to 20.

We know we're right.

We could also solve g of f of x equals 20.

An expression for g of f of x would be four lots of bracket x plus 8.

So if g of f of x equals 20, then four lots of x plus 8 must equal 20.

So x plus 8 must equal 5, and x must be equal to -3.

Again, let's check that that is the value that maps to 20 in the function g of f of x.

g of f of -3 is indeed equal to 20.

We know that we're right.

If f of x equals x minus 8, and g of x equals 3x plus 5, we could solve f of g of x equals 0, and g of f of x equals 26.

If f of g of x equals 0, we can express f of g of x as 3x minus 3.

3x minus 3 must be equal to 0 if f of g of x equals 0.

From there, add 3 to both sides.

Divide both sides by 3, x equals 1.

Let's evaluate f of g of 1, and we get 0, we know we're right.

g of f of x as an expression would be 3x minus 19.

So if we're solving g of f of x equals 26, then 3x minus 19 must be equal to 26.

3x must be equal to 45.

x must be equal to 15.

You know what we're gonna do now, don't you? We're gonna check that that is the value that maps to 26.

when it goes into the function g of f of x.

g of f of 15 is indeed equal to 26.

We know that's right.

We can also solve the equation f of x equals g of f of x.

f of x we know to be x minus 8.

g of f of x we know to be 3x minus 19.

So to solve f of x equals g of f of x, we have to solve X minus 8 equals 3x minus 19.

A few manipulations from there, and we arrive at the solution x equals 5.

5, or x equals 11 over 2, if you're communicating that answer as a fraction.

To check that this one's right, we'll substitute that value 11 over 2 into the function f, and we get -5 over 2.

We substitute it into the function g of f, and we get -5 over 2.

We know that we're right.

The solution to our equation is the moment where both functions map the same value in the domain to the same value in the range.

Both functions map the input 11 over 2 to the output -5 over 2.

Does this remind you of anything that you've seen before in mathematics? Simultaneous equations, I imagine you said.

Quick check you can solve a few equations like that.

f of x equals 2x, g of x equals 5x minus 3.

I've also given you expressions for the functions f of g of x and g of f of x.

Try to use all those expressions to solve three equations, f of g of x equals 114, g of f of x equals -83, and finally, f of x equals g of f of x.

Pause, have a go at those three.

And I'll see you in a few moments to check that your work matches mine.

Welcome back.

Let's see how we did.

First, solving f of g of x equals 114.

Our expression for f of g of x was 10x minus 6.

When that's equal to 114, 10x is equal to 120, x is equal to 12.

Is 12 the input that's going to map to 114 when it goes into the function f of g of x? It is indeed.

We know that's right.

Solving g of f of x equals -83.

Well g of f of x we know is 10x minus 3.

So when does 10x minus 3 equal -83? It's when 10x equals -80, and x equals -8.

Again, let's show that -8 maps to -83.

It does, we know that's right.

Lastly, f of x equals g of f of x.

A few manipulations from there, and we find that x equals 3 over 8.

Let's put that into both functions.

When we evaluate f of 3 over 8, we get 3/4.

Evaluate g of f of 3 over 8, and we get 3/4.

We know we're right.

Aisha and Laura are working with the exact same functions we've just seen.

f of x is still 2x, g of x is still 5x minus 3, which means f of g of x hasn't changed, and g of f of x hasn't changed.

Laura says, "I'm great at these.

I can solve any equation involving composite functions." Love your confidence, Laura.

"I'm going to solve f of g of x equals g of f of x next." But Aisha says, "I'm not so sure, I don't think anyone can solve that one." What do you think? With whom do you agree? Welcome back.

I wonder if you spotted it.

fg of x equals gf of x.

Well, to solve that, we're going to have to solve the equation 10x minus 6 equals 10x minus 3.

If we rearrange that, we end up with 0x equals 3.

I.

e.

, the value of x multiplied by 0 equals 3.

Laura's rightly confused.

"Oh no, I can't solve this.

Where did I go wrong?" Aisha says, "You didn't.

You've shown that there won't always be a solution." Not all equations have solutions.

That fact remains true for equations involving functions.

Here I've graphed f of g of x and g of f of x.

Graphing the two functions, we see that they are parallel linear graphs.

They have no intersection, therefore there is no solution to f of g of x equals g of f of x.

Okay, let's check you've got that.

Which of the below can be solved, which cannot? I've given you two functions, f of x and g of x, and I've given you the two composite functions, f of g of x and g of f of x.

There's 4 equations there, A, B, C, and D.

Some of them can be solved, some of them can't.

Pause, have a think, see if you can figure out which is which.

Welcome back.

Hopefully you said A can be solved and B can be solved.

If we're solving f of x equals g of x, then we're solving -4x equals three lots of x minus 1, which we can absolutely solve.

C, we can solve f of x equals f of g of x, well, that would be when -4x equals -12x plus 12.

We can solve that one.

A, B, and C are all linear equations with one solution.

D however, cannot be solved.

If you try and find a solution to f of g of x and g of f of x, you're trying to solve -12x plus 12 equals -12x minus 3.

If you rearrange that, we'll come to that same problem we saw previously.

x multiplied by 0 makes 15.

There is no x value that can make that true.

There are no solutions to equation D, f of g of x equals g of f of x.

We can solve a variety of other equations, too.

Here's three functions, f of x, g of x, and h of x.

We can solve equations such as 2f of x equals g of x, and f of x equals g of x plus h of x.

Let's have a look at how we do these.

In the case of two lots of f of x, well, that's two lots of our function of x, two lots of 2x minus 1.

On the right hand side we've got g of x, where g is just five lots of x minus 7.

From here we'll expand both the brackets on each side, and we'll rearrange from there to get x equals 33.

Again, it's useful to check our work.

If we evaluate 2f of 33 we get 130.

If we evaluate g of 33, we get 130, so we know we're right.

Both functions map 33 to 130.

In the case of g of x plus h of x, we have to be careful when we see that, this is not a composite function, it's not the output of h of x going into the function g of x.

This is the combined outputs of g of x and h of x.

So we'll set up this equation.

On the left hand side we've got f of x, which is 2x minus 1, g of x plus h of x, well, that's our function, g of x, five lots of x minus 7, and add our function h of x, which is -4x.

We can expand and simplify on the right hand side, and turn that into x minus 35 there.

From here, rearrangement takes us to our solution, x equals -34.

Again, we can check that solution.

Let's evaluate f of -34, that's -69.

And let's evaluate g of -34 plus h of -34.

That's -69, so we know we're right.

Quick check that you can do that, that you can solve two similar-looking equations.

Here's three functions for you, f of x, g of x, and h of x.

I'd like to use those to solve two equations.

I'd like to solve 5f of x equals g of x, and 4f of x plus h of x equals g of x.

Pause, give these a go now, right back in a moment to check your work.

Welcome back.

Firstly, let's solve 5f of x equals g of x.

Five lots of f of x will be five lots of 2x minus 1.

On the right side, we've got the function g of x.

Did you see the shortcut we can take from here? We can divide both sides by 5.

When we rearrange from here, we find that x equals -6, and of course we're going to check that.

We'll evaluate five lots of f of -6, which is -65, and g lots of -6, which is also -65.

We know we're right, both functions map -6 to -65.

Let's solve four lots of f of x plus h of x equals g of x now.

Four lots of f of x will be four lots of 2x minus 1.

Add h of x, which is -4x, and that will equal g of x, which is five lots of x minus 7.

We've got some expanding, simplifying, and rearranging to do from here, which will look like that.

Once we're in this position, we find that x equals -31, and you know we're going to check that.

So g of 31 is 120, hopefully four lots of f of 31 plus h of 31 is also 120.

It is, both functions map 31 to 120.

Solving f of x equals f inverse of x or g of x equals g inverse of x gives us an interesting result.

No spoilers here, I'm not gonna tell you what's happening, you're gonna tell me what's happening.

f of x equals 6x minus 15, so f inverse of x is x plus 15 over 6.

So let's solve f of x equals f inverse of x.

6x minus 15 equals x plus 15 over 6.

Let's multiply both sides by 6.

Rearrange to find 35x equals 105, x equals 3.

Let's check that we're correct.

f of 3 is indeed 3.

f inverse of 3 is indeed 3.

We know we're right.

Over to you now.

I'd like you to solve g of x equals g inverse of x.

And then once you've solved that, here's a key question for you.

What do you notice about the solutions to g of x equals g inverse of x, and f of x equals f inverse of x? there's something rather special about the solutions, and you're going to spot it.

Pause, and do that now.

Welcome back.

Let's see how we did.

Firstly, let's solve the equation.

Four lots of x minus 9 equals x over 4 plus 9.

Let's expand that bracket on the left hand side.

Let's multiply all the terms by 4, and let's rearrange and solve x equals 12.

If we're right, then g of 12 is equal to g inverse of 12, and it is, we know we're right.

Next, I asked you to spot what's interesting about the solutions to our two equations.

You might have said in both cases the input and the outputs are the same value.

Did you spot that? Well done.

Let's have a look at what's going on here.

In the case of f of x and f inverse of x, and g of x and g inverse of x, a function and its inverse are symmetrical in the line y equals x.

The intersection of the two functions' graphs will always be on the line y equals x because of that symmetry.

Hence in our solutions, the input and output were the same value.

How wonderful, eh? If f of x equals x plus 7, f inverse of x equals x minus 7.

Let's solve f of x equals f inverse of x.

x plus 7 equals x minus 7, 7 plus 7 equals x minus x.

Hmm, something's going wrong here.

14 equals 0x.

x multiplied by 0 makes 14.

We have a problem, there's no solution.

What's happened here you can see in this graph.

A function and its inverse are symmetrical in the line y equals x.

If the graph of the function f of x does not intersect y equals x, then f of x equals f inverse of x will have no solutions.

Quick check you've got that.

True or false, f of x equals f inverse of x will always have a solution? Is that statement true? Is it false? Once you've decided, use one of the two statements at the bottom of the screen to justify your answer.

Pause and have a think about this.

Welcome back.

I hope you said false.

And then justified that with f of x equals f inverse of x will only have a solution if the graph of f of x intersects the line y equals x.

We will encounter quadratics when solving with composite functions.

f of x equals 2x minus 1, g of x equals x squared.

fg of x is gonna equal 2x squared minus 1, gf of x is gonna equal 2x minus 1 squared.

Let's solve f of g of x equals 49.

In that case, 2x squared minus 1 will equal 49, and Laura says, "Ooh, I like these, I'll solve this one." Go on then, Laura, 2x squared equals 50, x squared equals 25 x equals 5.

Laura checks her work.

fg of 5 equals 49, and Laura says "It works.

My solution is right." But Aisha says "Yes, that solution works, but you missed something." Can you see what Aisha has spotted? Welcome back.

I wonder if you spotted it.

Laura spotted it.

Laura's reflected on her work and said, "Of course, I need to take the positive and negative roots." From x squared equals 25, we've got two solutions, x equals positive or negative 5.

Let's check that.

fg of 5 equals 49, and fg of -5 equals 49.

Aisha says, "Well done Laura, you found both solutions." I say well done, both of you.

Remember to take both the positive and negative roots when solving a quadratic equation.

Quick check you can do that now.

I solved fg of x equals 49.

I'd like you to solve gf of x equals 49.

Pause and do that now.

Welcome back.

Let's see how we did.

If gf of x equals 49, then 2x minus 1 squared equals 49.

Take both the positive and negative roots, and then rearrange and we find two solutions, x equals 4 and x equals -3.

We can check those.

gf of 4 equals 49, and gf of -3 equals 49.

You know you're right.

Practise time now.

There's two functions, f of x and g of x, and I've given you the composite functions f of g of x and g of f of x.

I'd like you to solve A, B, and C, and I'd like you to show in part D that that equation has no solutions.

Pause, give these four problems a go now.

Question 2.

There's three functions, f of x, g of x and h of x, and I've given you f inverse of x, and the composite functions gh of x and hf of x.

There's four equations there that I'd like you to have a go at solving, pause and try them now.

Right, let's see how we did.

Question one.

Part A, solving fg of x equals 113, 40x minus 7 will equal 113, so 40x must equal 120, so x must equal 3.

For part B, we should have written five lots of 8x minus 7 equals -235, and then we can divide both sides 3 by 5, and find that x equals -5.

For C, g of x equals gf of x, would look like so.

Lots of roots we can take from there, dividing both sides by 5 is a very efficient one, so 7x must equal 7, x must equal 1.

For part D, I actually should show that fg of x and gf of x has no solutions.

You'd start by writing that, expand that bracket.

We can see now we've got no solutions, because 0x equals -28, or you might say when I graph both functions, they're parallel lines, therefore there's no intersection, no solution.

Question 2 part A, f of x equals f inverse of x.

Our solution to that would look like so.

We find that x equals -1.

f of -1 equals -1, f inverse of -1 equals -1.

Inputs and outputs match if a function and it's inverse have a solution.

For part B, gh of x equals 717.

This is going to be a quadratic.

5x squared is gonna equal 720, x squared's gonna equal 144.

Don't forget to take both the positive and negative roots there.

x equals positive 12 and x equals -12.

Question 2, part C, hf of x equals 225.

We set up like that.

We take the positive and negative square roots, and we find x equals -11 over 4 and x equals 1.

We have two solutions to that one.

For part d, g of x plus h of x looks like that on the left hand side.

It a bit of rearranging to happen there to reach this quadratic, which factorises, giving us two solutions, x equals -2 and x equals 5.

Again, two solutions to that equation.

Onto the second half of our learning now where we're gonna look at solving using multiple methods.

f of x equals x minus 4, g of x equals 7x.

Solve gf of x equals 84.

In all the examples we've seen so far, we've been given the expression GF of X.

Do we need it in order to solve this equation? There was the answer, we certainly can solve the equation if we know gf of x.

gf of x equals seven lots of x minus 4.

When does that equal 84? It equals 84 when x equals 16.

This works, but did we need that expression gf of x? Not necessarily.

gf of x equals 84 means the output of f of x is input into the function g of x in order to generate the value 84.

So we could start solving this by asking what output of g of x gives us 84? g of x equals 84.

Well, 7x must equal 84, so x must equal 12.

So 12 must be the output of f of x.

If gf of x equals 84, and g of 12 equals 84, then f of x equals 12.

You can see it when we write g of 12 equals 84, and g of f of x equals 84, f of x must be 12.

So let's solve f of x equals 12.

x minus 4 equals 12, x must equal 16.

Let's check that's right.

Let's evaluate gf of 16.

gf of 16 becomes g of 12, which is 84.

We know we're right.

Our check moves forward through gf of x to 84.

Our method essentially moved backwards through it from 84 back to 16.

Comparing these two methods is interesting.

On left hand side we've got method one, knowing the function gf of x.

On the right hand side we've got method two, using inputs and outputs.

When we look at what's happened during these two methods, we've got common steps.

On both sides we've done divide by 7, that's the inverse of g of x.

We've also taken the step somewhere of plus 4 to both sides.

That is the inverse of f of x.

Essentially the same thing happening in both methods, they just look different.

It's personal preference as to which method you use.

Sometimes one method will be clearly more efficient than the other.

Quick check you've got that.

True or false.

If f of x equals -2x, and g of x equals x plus 13, we must work out the function fg of x in order to solve fg of x equals 20.

Is that true or is it false? Once you've decided, use one of the statements at the bottom of the screen to justify your answer.

Welcome back.

I hope you said false.

I hope you justified that with you can solve by considering inputs and outputs of f of x and g of x without ever knowing the function f of g of x.

Next I'm gonna solve an equation using two different methods and then I'll ask you to do something similar.

I'm gonna solve fg of x equals 20.

I'm gonna use two methods, first of which is knowing the function fg of x, second of which is operating with inputs and outputs.

Knowing the function fg of x will look like so.

I'm gonna set up the equation, <v ->2 lots of x plus 13 equals 20.

</v> divide both sides by -2 and I find that x equals -23.

In terms of inputs and outputs, if f of g of x equals 20, then I need to know what is it that's going into the function of x to make 20.

Well, that's gonna be -10, so that means g of x must be equal to -10.

When does that happen? Well, that happens when x equals -23.

No surprise that I found the same result using a second method.

Your turn now.

I'd like you to solve g of f of x equals 17 using both methods.

You're using the same functions f of x equals -2x, and g of x equals x plus 13.

Pause, give this problem a go now.

Welcome back.

Method one knowing the function.

That would look like so, g of f of x is -2x plus 13.

When does that equal 17? It equals 17 when -2x equals 4, or x equals -2.

Method two, inputs and outputs.

g of what makes 17? If we can find that what we'll know what f of x equals.

If g of x equals 17, then x equals 4.

So if f of x equals 4, the whole thing's going to work.

When does f of x equal 4? When x equals -2, absolutely no surprise to get the same answer using a different method.

In fact, it's very reassuring.

Aisha and Laura are solving this equation.

gf of x equals 48.

Aisha says, "I'm always gonna use the inputs outputs method every time." f of x equals 48.

4 equals 48, x equals 12.

When does g of x equal 12? That occurs when x equals 10.

Laura says, "Let's check that solution." g of f of 10 equals 72.

Something has gone wrong.

Can you see what's gone wrong in the maths here? Pause, see if you can spot it.

Welcome back.

Did you notice? "We've put the functions in the wrong order," Laura says.

"You're right.

It's the output of f of x that becomes the input of g of x." You might think of it like this, g of something equals 48, and that something is f of x.

g of what makes 48? That's 28, so f of x must equal 28.

So 4x equals 28, x equals 7.

And when we go to check that, we find that gf of 7 is indeed 48, we know we're right.

Quick check you've got that.

Which of these two methods is correct for fg of x equals 44? Is it A or is it B? Pause, see if you can spot it.

Welcome back.

Let's see how you did.

Hopefully you said it's option A.

Practise time now.

Question one, f of x equals 9x, g of x equals 8x minus 1.

Part A, I'd like you to solve f of g of x equals 495, and do that twice using two different methods.

For part B, g of f of x equals -225, again twice using two different methods.

Pause, give these a go now.

Question 2, three functions, f of x, g of x, h of x, and there's four equations for you to solve.

Pause and enjoy.

Feedback time.

Let's see how we did.

Solving f of g of x equals 495 twice using two different methods.

Method one, knowing the function.

Method two using inputs and outputs.

If we know the function of fg of x, then we can set up that equation.

From there, there's lots of ways we could solve it.

Whichever way you solve it, you arrive at x equals 7.

Method two, using inputs and outputs.

f of x must equal 495 f of what makes 495, 55, so g of x must equal 55.

if g of x equals 55, then we get to x equals 7.

In mathematics, using two different methods and arriving at the same answer reassures us we are correct.

For part B, gf of x equals -225.

Knowing the function would look like so.

And you find that x equals -28 over 9.

Using inputs and outputs, g of something makes -225.

That something is -28.

f of x equals -28, so x equals -28 over 9.

And here's Aisha to say "Two different methods plus the same answer equals confidence." That's right, Aisha.

Question 2, part A should have looked like so.

x equals 5.

Part B, your method should have looked something like that.

x equals 1.

For part C, we find that x equals -2, and for part D we find that x equals 0.

Well, we're at the end of the lesson now, sadly.

Well, what have we learned? We've learned that composite functions can lead to equations which may be solvable.

There's a choice of methods when solving an equation, like g of f of x equals 196.

An expression for the composite g of f of x can be found and solved to find x.

Alternatively, we can consider inputs and outputs.

Hope you enjoyed this lesson as much as I did, and I'll look forward to seeing you again soon for more mathematics.

Goodbye for now.