Lesson video

Hello, Mr. Robson here.

Welcome to maths, what a lovely place to be.

Today, we're defining function notation.

There's loads of mathematical efficiencies in this lesson, so let's not hang around, let's get stuck in.

Our learning outcome is that we'll be able to define function notation.

Lots of keywords you're going to hear in our learning today.

Function, domain, 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 a set of values that the mapping is performed on.

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

Look out for those words throughout our learning today.

Two halves to the lesson, we might going to begin by looking at f of x.

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

A function like times 3 plus 5 uniquely maps an input to an output.

All of the inputs make up the domain and all of the possible outputs make up the range.

We can reflect all that in this table of values.

Those values in the domain are mapped onto those values in the range.

And if we perform this function on the input of x, we get 3x plus 5.

This is called the function 3x plus 5, but we write it using the shorthand notation.

f of x equals 3x plus 5.

Quick check you've got that.

Can you match the descriptions to the function notation? On the left-hand side, we have four descriptions of functions.

On the right-hand side, we have four examples of function notation.

Can you correctly pair them up? Pause and give it a go now.

Welcome back.

Let's see how we did.

Hopefully you paired the function 4x minus 9 to f of x equals 4x minus 9.

And you paired x is multiplied by 4 and then 9 is added to f of x equals 4x plus 9.

Hopefully you paired the function 9 minus x squared to f of x equals 9 minus x squared.

And finally, if x is squared and then 9 is added, we write that far more concisely if we wrote f of x equals x squared plus 9.

On the right-hand side here, you can see how we efficiently write these functions.

Andeep and Sam are discussing function notation.

Andeep says, in algebra, ab is the simplification of a multiplied by b.

Doesn't that mean that f of x is the same as f multiplied by x? Andeep's not sure.

He's asking the question.

What do you think? I wonder what you thought.

Sam knows.

Sam's got an outstanding answer.

Sam says, in algebraic notation, ab is indeed a multiplied by b, but f of x is function notation.

F of x means a function of x.

Be careful about this, In f of x, x is still a variable, but the f is not a variable.

Quick check you've got that.

Which of the below are true statements? Some are true, some are false, pause and pick them out.

Welcome back.

Hopefully you said A is false.

F of x does not equal f multiplied by x.

Hopefully you said B was true.

F of x does indeed mean a function of x.

C was false.

F of x certainly does not mean f equals x.

And D was true.

F of x uniquely maps values from the domain to the range.

Graphing functions requires the same set of skills that we use when graphing equations.

If they ask you to draw the graph of the equation y equals 3x plus 5, you might well start with a table of values.

You'll have your x values which you input into the equation to generate your y values.

You then have coordinate pairs which you plot.

And it's a linear equation, so you plot a linear graph.

Very similar thing happens when we plot f of x equals 3x plus 5.

Those x values are input into the function, we have those outputs, we plot the coordinate pairs, and draw a line.

Now we could label the axis like this, but it's not very efficient.

It's more efficient if we label the axis like this.

When you draw linear graphs, you often have your axis labelled as x and y.

When we're drawing functions, you might see those axis labelled as x on the horizontal axis and f of x on the vertical axis.

The difference between the two is what defines a function.

On the left-hand side, when we graph y equals 1 over x, that is the graph of that equation.

And that's in the table of values.

There's no output when x equals zero because y equals 1 over zero, that's undefined.

So in the graph, the y-axis is an asymptote, and that is the graph y equals 1 over x.

It's a problem though when we try to graph f of x equals 1 over x.

This is not a valid function.

Valid function uniquely maps all inputs onto a value in the range.

But zero is not a valid input, so that's not correct.

It's not a valid function.

We can make it a valid function by restricting the domain instead of just saying we're graphing f of x equals 1 over x, we say we're graphing f of x equals 1 over x, but x cannot be equal to zero.

This is now a valid function because all values in the domain can be uniquely mapped onto the respective values in the range.

Let's have a quick check to see if you've got that.

Which of the below is wrong and why? There's some maths in the rectangle on the left-hand side of your screen and some maths on the right-hand side.

One of them is wrong.

Can you spot which one is wrong and can you explain why? Pause, have a good think about this problem now.

Welcome back.

I wonder if you spotted it.

I wonder what you said.

Hopefully you said left-hand side's perfectly fine.

Y equals 2 over x minus 1, well that's perfectly correct.

That is the graph of that equation.

On the right-hand side we have a problem.

It's not a valid function unless we restrict the domain.

We'd have to add the domain restriction x cannot be equal to 1, because that denominator cannot be zero.

You'll still see the y-axis labelled as y when graphing functions.

This is different.

A moment ago I said the vertical axis might be labelled as f of x.

It might, but you'll also see it labelled as y when we're graphing functions.

When this happens, you'll see this, y equals f of x.

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

In this case, x values are being uniquely mapped to y values.

Now why would we want to write it all like this? Do pardon the pun.

It's particularly useful when we see multiple functions graphed on the same axis.

There we've graphed the function 3x plus 5.

What if we wanted to graph the function g of x equals 7 minus 2x.

We've got two functions here.

One is f of x, the function f.

Another is g of x, the function g.

You might also see functions called h of x, i of x, j of x.

This happens when we want to map multiple functions.

When we're mapping the g of x function, we'll write y equals g of x.

And there we have it on the same graph, the graph of f of x and the graph of g of x.

When substituting into functions, use this notation.

F of negative 2, we write the function f of x equals 9 minus 3x, and x is variable.

We need to substitute values in in order to graph this function.

So when I write f of negative 2, I've substituted any value of negative 2, so I put that into the function.

9 minus 3x becomes 9 minus 3 lots of negative 2.

We can evaluate that.

That becomes 15.

The f of negative 1, that's 9 minus 3 lots of negative 1.

We can evaluate that, it's 12.

F of zero.

When x equals zero, the function has a value of 9.

And we can keep going and evaluate f of 1, f of 2, f of 3.

This is showing that we've substituted in the x values negative 2, negative 1, 0, 1, 2, 3.

And we have those corresponding values, the outputs.

We can plot those coordinate pairs and draw a line through them.

Over to you now.

It's your turn.

I just populated the table of values for y equals f of x.

What I'd like you to do is complete these substitutions and populate the table for the function g of x.

I've started what you need to write for g of negative 2 and g of negative 1.

The rest of those you're gonna have to write yourself.

Pause and do this now.

Welcome back.

Hopefully your substitutions looked like so.

G of negative 2 evaluates to negative 6.

G of negative 1 evaluates to negative 2.

G of zero is 2, g of 1 is 6, g of 2 is 10, g of 3 is 14.

Your table values looks like so.

And I'll plot that graph for you.

Next check.

We've got three functions here.

F of x is x squared plus 5, g of x is x squared plus 5x, and the function h of x is x plus 5 squared.

What I'd like you to do is match the notation with the correct value f of 4.

If 4 goes into our function f of x, what value comes out? G of 4.

If 4 goes into our function g, what value comes out? I'd like you to pause and pair those up now.

Welcome back.

Hopefully you paired f of 4 with 21 because when 4 goes into the function f of x, we get 4 squared plus 5, which is 21.

G of 4, you should have paired with 36, because g of 4 becomes 4 squared plus 5 lots of 4, which is 36.

H of 4, you should have paired with 81.

F of negative 3, you should have paired with 14.

G of negative 3 with negative 6.

And H of negative 3 with 4.

Practise time now.

Question one, I'd like you to evaluate the following.

We've got three functions.

F of x is 5x minus 12, g of x is x squared plus 5x, and h of x is x cubed minus x squared.

Can you evaluate all nine of those? Pause and do that now.

For question two, Sam is mapping the function f of x equals 15 minus 7x.

And Sam says, I know that f of 2 equals 1, so I think f of negative 2 will equal negative 1.

It must be the opposite.

What I'd like you to do is show Sam why this thinking is wrong.

Pause and do that now.

Feedback time.

Question one I ask you to evaluate the following.

For A, g of zero.

Well, if zero goes into the function g, zero comes out.

For B, we're evaluating h of 1.

If 1 goes into the function h, we get 1 cubed minus 1 squared, which is also zero.

For C, f of 2, 2 goes into the function f, we get 5 lots of 2 minus 12 that's negative 2.

For part D, h of negative 3, negative 3 goes into the function h, we get negative 3 cubed minus negative 3 squared, that gives us negative 36.

G of negative 6, we should evaluate to 6.

F of negative 9 evaluates to negative 57.

You might want to pause, check that your method and your answers match mine.

Let's do parts G, H and I now.

Notice these are not integrate inputs, but we can input any value into these functions.

So f of 0.

5, that evaluates to negative 9.

5.

Evaluating h of 9/4 or 9 over 4.

Well if 9 over 4 goes into the function h, we get 5 over 4 cubed minus 9 over 4 squared, which as a fraction in its simplest form will be 405 over 64.

Last one I, g of root 7, when root 7 goes into g of x, we will get root 7 squared plus 5 lots of root 7, that'll give us 7 plus 5 lots of root 7.

We weren't specified as to whether we should round that to any particular number of decimal places or significant figures, so I've just left that answer exact as 7 plus 5 root 7.

Question two.

That's a common misunderstanding this that Sam had.

Sam's mapping the function f of x equals 15 minus 7x.

And if f of 2 equals 1, then f of negative 2 equals negative 1 surely.

But I ask you to show Sam why this thinking is wrong and hopefully you evaluated f of negative 2.

If negative 2 goes into that function out comes 29.

So we can say, Sam, it's always a good idea to test your thinking.

There's now hypothesis that f of negative 2 might be negative 1, but it's useful to test it.

Thank you, says Sam.

You're welcome, we would say.

Onto the second half of our learning now.

Beyond f of x, this sounds interesting.

Let's take a look.

Function notation can be useful when we wish to manipulate a function.

We can evaluate the below for any valid value of x.

F of x, 2f of x, f of 4x, and f of x plus 5.

Well, what do they mean? We know what f of x means.

It means to evaluate the function for the given value of x.

What about 2f of x? This means evaluate the function for the given value of x, then multiply the result by 2.

F of 4x, that means multiply the given value of x by 4, then substitute this value into the function.

It's important that you multiply x by 4 before performing the function on it.

F of x plus 5, well we're gonna add 5 to the given value of x and then substitute that into the function.

Let's check you've got that.

I'd like to match the below function notation to the correct meaning.

Five bits of function notation, five different descriptions, pause and pair them up.

Welcome back.

Hopefully you paired f of x with the simplest explanation, evaluating the function for the given value of x.

F of x minus 8, you should have paired with subtract 8 from the given value of x, then substitute this value into the function.

F of 8x, that means multiply the given value of x by 8, then substitute this value into the function.

F of x plus 8, we're adding 8 to the given value of x, then substituting this value into the function.

Leaving us with 8f of x, which is to evaluate the function for the given value of x, then multiply the result by 8 We can evaluate each of these things when x equals 8.

If we know that f of x equals 3x minus 10, then we can evaluate f of 8.

We'll put 8 into our function.

3 lots of 8 minus 10 gives us 14.

If x equals 8, we can evaluate 2 lots of f of x.

It's 2 lots of f of 8.

We'd write that as 2 lots of f of 8, and oh, wonderful, we know the value of f of 8, it's 14.

So we need 2 lots of 14, that's 28.

F of 4x is a little trickier.

It's f of 4 lots of 8, because x is 8, so we need to find f of 32, which is 3 lots of 32 minus 10, that's 86.

F of x plus 5, well that's f of 8 plus 5 because x equals 8.

F of 13 becomes 3 lots of 13 minus 10, that's 29.

One for you to do now.

If g of x equals 7 plus 3, I'd like you to evaluate each of those when x equals 5.

Pause and do that now.

Welcome back.

G of x, nice and straightforward.

7 lots of 5 plus 3, that's 38.

5 lots of g of x, well that's 5 lots of g of 5.

And we know what g of 5 is, it's 38.

So 5 lots of 38, that's 190.

G of 2 of x, well that's g of 2 lots of 5 or g of 10.

If 10 goes into the function g of x, we get 7 lots of 10 plus 3, that's 73.

G of x minus 1, that'll be g of 4, 7 lots of 4 plus 3, that's 31.

You might wanna pause now and just check that your method and your answers are the same as mine.

Andeep and Sam are discussing this pair of functions and this problem, f of x is 2x plus 7, g of x is 3x plus 2.

And Andeep and Sam are asked to evaluate f of 4 plus g of 5.

Andeep says, in algebra, we should always try to simplify.

That's a good idea, Andeep.

So I'm going to try and add f of x to g of x 2x plus 7 plus 3x plus 2 becomes 5x plus 9 before I substitute.

Sam says, hold on, I've spotted a problem Andeep.

Can you spot the same problem that Sam has spotted? Pause, have a conversation with a person next to you, or a good think to yourself.

I'll see you in a moment for the answer.

Welcome back.

Did you spot it? Sam says, what x value would we substitute in? 4 or 5? If we're being asked to evaluate f of 4, well that's x equals 4.

If we're asked to evaluate g of 5, that's x equals 5.

So Sam has a really good point here.

What are we going to substitute in to this 5x plus 9? In a case like this, we can't combine the functions, because we have two different x values to substitute in.

We would have to evaluate f of 4 and g of 5 separately and then sum the results.

F of 4 becomes 2 lots of 4 plus 7, that's 15.

G of 5 becomes 4 lots of 5 plus 2, that's 17.

So f of 4 plus g of 5 is 15 plus 17, that's 32.

Your turn to do the work now.

If f of x equals 11x plus 1 and g of x equals 17 minus 3x, I'd like you to evaluate f of 3 plus g of negative 3.

Pause and give this problem a go now.

Welcome back.

Hopefully you evaluated f of 3 to be 34, g of negative 3 to be 26, so f of 3 plus g of negative 3 becomes 34 plus 26, that's 60.

Practise time now.

Question one, f of x equals 7x plus 12, can you evaluate each of those when x equals 6.

Five problems to do there, pause and give them a go now.

Question two, we've got three functions to work with now.

F of x equals 7x plus 12, g of x equals 8 minus 7x, and h of x equals x squared minus 2x.

With that in mind, I'd like you to evaluate the following.

Five problems to evaluate there, pause and try them now.

Question three, f of x equals 3x plus 9, g of x equals 8 minus x.

For part A, it says evaluate f of 2 plus g of 2, and there's some math there and there's a tick there.

Part B says, evaluate f of 3 plus g of 4, and there's some math there and there's some crosses there.

What I'd like you to do is write at least one sentence to explain why this method obtains the correct answer in part A, but it does not obtain the correct answer in part B.

Pause and write at least one sentence now.

Welcome back.

Feedback time.

Question one, we were asked to evaluate when x equals 6.

For part A, f of 6 became 54.

For part B, 5 lots of f of 6 became 270.

For part C, f of 4x evaluated to 180.

Part D, f of x plus 3 became 75.

And for part E, 2 lots of f of x minus 1 became 94.

Now I'm gonna pause, check your method and your answers match mine.

For part two, again, we are evaluating.

Part A, f of 1 plus g of 2 became 13.

For part B, g of 3 plus h of 4 became negative 5.

Part C, f of 5 plus h of 6 became 71.

Part D, evaluated to 56.

Part E, evaluated to 118.

Again, pause and check that your method and your answers match mine.

For part three, I ask you to write a sentence explaining why we get the correct answer in A, but the method doesn't work in B.

In part A, x has the same value in both functions.

We're evaluating f of 2 and a g of 2, x equals 2 in both cases, so it's only one input when we add the two expressions together.

In part B, x does not have the same value in both functions, so we would need to evaluate each function separately.

We're at the end of the lesson now, sadly.

Well, we've seen some lovely maths, haven't we? We've learned that f of x means a function of x and not f multiplied by x.

We've learned that an efficient way to write x is multiplied by 7 then add 3 is to use function notation like f of x equals 7x plus 3.

We know that multiple functions can be written like this using f of x, g of x, h of x, and so on.

And we've learned that functions can be manipulated and evaluated.

For example, f of 2, g of 5, h of negative 1, 6 lots of f of 2.

Well, I hope you enjoyed this lesson as much as I did.

And I look forward to seeing you again soon for more maths.

Goodbye for now.