### Video Transcript

In this video, weโll learn how to
identify, write, and evaluate a linear function and complete a function table.

A linear function is an algebraic
equation that gives the graph of a straight line. Each term in the equation is either
a constant or a product of a constant and a variable like ๐ฅ. But the variables will never be
associated with exponents or powers. Each input to the function, which
we can treat like a function machine, will have exactly one output. These are all examples of linear
functions.

Now, this first bit is pronounced
๐ of ๐ฅ. ๐ is the name of the function, and
the ๐ฅ is what goes into it; itโs its input. In this function, for example, we
put ๐ฅ in, multiply it by two, and then subtract one. We could replace the ๐ฅ with a
number, such as five, and weโd perform the same set of operations. Our output will be different
depending on the number that went in.

Now, we might even come across
linear functions that are not given in standard form. Thatโs ๐ of ๐ฅ is equal to ๐๐ฅ
plus ๐ for real constants ๐ and ๐. Letting ๐ฆ be equal to ๐ of ๐ฅ,
these might look like this: ๐ฆ plus ๐ฅ equals five or three ๐ฅ minus ๐ฆ plus seven
equals zero. These equations still correspond to
the graph of a straight line; theyโre still linear functions.

Letโs begin by looking at how we
can find outputs for very simple functions.

Complete the given function
table.

In this table, we have an
input. Itโs defined as ๐ฅ. But weโre also given a list of
inputs. We have nine, five, and 16. So, ๐ฅ is going to be equal to
nine. Then itโs going to be equal to
five. And then, itโs going to be equal to
16. Our job is to calculate the
outputs. Now, the output is given as two
plus ๐ฅ. This is called a function. And it tells us what to do to our
input. In this case, itโs telling us to
take the number two and add ๐ฅ, which is the input, to it.

So, when ๐ฅ is equal to nine, we
replace โ remember, the mathematical word for this is โsubstituteโ โ ๐ฅ with
nine. In this case then, our output
becomes two plus nine, which is equal to 11. And so, the value of our first
output is 11. For our second output, weโre going
to input ๐ฅ is equal to five. And so, our output becomes two plus
five, which is equal to seven.

We have one more output to
calculate. And that is when ๐ฅ is equal to
16. In this case, our sum becomes two
plus 16, which is equal to 18. And so, we have the three outputs
for our function two plus ๐ฅ. Theyโre equal to 11, 7, and 18. Now, this is, in fact, an example
of a linear function. Letโs define our output as being
equal to ๐ฆ. So, ๐ฆ is equal to two plus ๐ฅ. Then, our pairs of values
correspond to coordinates. When ๐ฅ is equal to nine, ๐ฆ is
equal to 11. We have another coordinate five,
seven and a third coordinate 16, 18. Plotting these on the coordinate
plane might look a little something like this. Our points lie on a straight
line. And this is how we know we have a
linear function.

In our next example, weโll look at
an alternative way that we could present a function.

Find the value of ๐ of eight given
the function ๐ of ๐ฅ equals three minus seven ๐ฅ.

In this question, weโve been given
a function whose name is ๐. The ๐ฅ is what goes into the
function; itโs its input. Weโre looking to find the value of
๐ of eight. Remember, the bit inside the
parentheses or brackets is the input. So, what weโre really trying to
find is whatโs the output when the input is eight. So, we look at our function ๐ of
๐ฅ equals three minus seven ๐ฅ. And each time we see an ๐ฅ, we
replace or we substitute it with eight.

So, ๐ of eight is equal to three
minus seven brackets eight. Now, be careful. A common mistake is to write 78
here. But remember, seven ๐ฅ means seven
multiplied by ๐ฅ. So, this bit is actually seven
times eight.

PEMDAS or BIDMAS, thatโs the order
in which we perform the various operations, tells us to multiply before performing
any subtraction. So, we work out seven multiplied by
eight first, which is, of course, 56. ๐ of eight is, therefore, three
minus 56. Now, donโt change the order of the
sum. Weโre not calculating 56 minus
three. And three minus 56 is negative
53. So, given the function ๐ of ๐ฅ
equals three minus seven ๐ฅ, the value of ๐ of eight is negative 53.

Weโll now look at an inputโoutput
table for a similar type of function.

Fill in the inputโoutput table for
the function ๐ฆ equals five ๐ฅ plus three.

Our table gives us a row of
inputs. Theyโre zero, two, four, and
five. Our job is to calculate the outputs
given a specific function. The function we have is ๐ฆ equals
five ๐ฅ plus three. The input is ๐ฅ, and ๐ฆ is the
output. There might be times youโll see the
output denoted as ๐ of ๐ฅ. This is essentially the same
thing.

Now, what this function tells us is
that the output is calculated using the expression five ๐ฅ plus three. Itโs a function. And it tells us what to do to our
input. In this case, we take the ๐ฅ, the
input, multiply it by five, and then add three. So, letโs do this for each of our
inputs.

They are ๐ฅ equals zero, ๐ฅ equals
two, ๐ฅ equals four, and ๐ฅ equals five. The function is ๐ฆ equals five ๐ฅ
plus three. So, we begin by substituting ๐ฅ
equals zero. So, our function becomes ๐ฆ equals
five times zero plus three. Thatโs three. And so, the first output in our
table is three.

Letโs repeat this process for our
second input. This time, weโre going to replace
๐ฅ with two. Our function becomes ๐ฆ equals five
times two plus three, which is equal to 13. And so, the second output is
13.

Our third input is ๐ฅ equals
four. And so, we get ๐ฆ equals five times
four plus three, which is equal to 23. Weโll repeat this process one more
time. This time our input is five. And so, our output is found by
calculating five times five plus three, which is 28. Our outputs are then three, 13, 23,
and 28.

Now, letโs go back to the table in
this example. In the first three inputs, we
increase by two each time. And in the first three outputs, we
increase by 10 each time. This constant common difference
tells us that weโre likely working with a linear function. Remember, a linear function
produces the graph of a straight line, so this makes a lot of sense.

Weโll now look at how to find an
unknown given a linear function and an output.

Find the value of ๐ given ๐ of ๐ฅ
equals ๐๐ฅ plus 13 and ๐ of eight is equal to negative 11.

Here is our function. ๐ is sort of the name of the
function. And ๐ฅ is what goes in; itโs its
input. Weโve also been given some
information about the value of ๐ of eight. In other words, our input is no
longer ๐ฅ. Itโs eight. And when the input is eight, the
output is negative 11. So, letโs see what happens when we
put eight into our definition of the function here.

๐ of eight means each time we see
the ๐ฅ, we replace or substitute it with eight. So, ๐ of eight becomes ๐ times
eight plus 13. Letโs write that as eight ๐ plus
13. But of course, we know that ๐ of
eight is equal to negative 11. So, this must mean that eight ๐
plus 13 must be equal to negative 11. And so, our job, weโre trying find
the value of ๐, is to solve this equation.

Weโll do this by performing a
series of inverse operations. The first thing weโre going to do
is subtract 13 from both sides of our equation. Remember, eight ๐ plus 13 minus 13
is just eight ๐. And negative 11 minus 13 means we
move 13 spaces further down the number line. And we get to negative 24. So, eight ๐ is equal to negative
24.

The eight is multiplying the
๐. And so, the inverse operation we
apply next is to divide both sides of our equation by eight. That leaves ๐ on the left-hand
side. And since 24 divided by eight is
three, we know negative 24 divided by eight is negative three. And so, given ๐ of ๐ฅ is equal to
๐๐ฅ plus 13 and ๐ of eight is equal to negative 11, we find ๐ is equal to
negative three.

Now, we can check our answer by
letting ๐ of ๐ฅ now be equal to negative three ๐ฅ plus 13. Weโve replaced ๐ with negative
three. Weโre going to check that the value
of ๐ of eight is indeed negative 11. And so, we replace ๐ฅ with
eight. And we get negative three times
eight plus 13. Thatโs negative 24 plus 13, which
is negative 11 as required.

Weโll now look at how to work out a
function given some inputs and outputs of that function.

Find the rule for the given
function table.

In this question, we have some
information about some inputs. The general form of this input is
๐ฅ. And the three inputs weโve been
given are ๐ฅ equals one, ๐ฅ equals four, and ๐ฅ equals 10. So, letโs imagine weโre given a
function machine. When we take the input one, we get
an output of nine. When we take an input of four, we
get an output of 12. And when we input ๐ฅ equals 10, we
get an output of 18.

Weโre trying to find a general
rule. So, our job is to work out what we
get when we input ๐ฅ. So, what is happening each
time? We can actually observe that weโre
simply adding eight each time. One add eight is nine, four add
eight is 12, and 10 add eight is 18. The rule, then, is to add eight to
our input to get the output. But how do we define that
algebraically? Well, if the general input is ๐ฅ,
we add eight. And we simply write that as ๐ฅ plus
eight. And thatโs our output. Itโs ๐ฅ plus eight.

In our final example, weโll look at
how changing the input to an alternative algebraic expression will affect our
function.

Evaluate ๐ of four minus ๐ฅ, given
that ๐ of ๐ฅ is equal to three ๐ฅ plus seven.

What this notation is telling us is
that when we take a function named ๐ and we input ๐ฅ, our output is three ๐ฅ plus
seven. We can change our input, that is,
replace ๐ฅ with any number, and weโll get a different output. For example, ๐ of two, we replace
๐ฅ with two. And our expression becomes three
times two plus seven, which gives us 13.

But we havenโt been given a single
number. Weโre told to evaluate ๐ of four
minus ๐ฅ. And so, instead of replacing ๐ฅ
with a number like two, weโre actually just going to replace ๐ฅ with our input, with
four minus ๐ฅ. And so, ๐ of four minus ๐ฅ becomes
three times four minus ๐ฅ plus seven.

Letโs distribute the parentheses or
expand the brackets. To do so, weโre going to multiply
the three by the four and the three by the negative ๐ฅ. Three multiplied by four is 12. And three multiplied by ๐ฅ is three
๐ฅ. So, three multiplied by negative ๐ฅ
is negative three ๐ฅ. ๐ of four minus ๐ฅ is, therefore,
12 minus three ๐ฅ plus seven. We then collect like terms. 12 plus seven is 19. And so, we see that ๐ of four
minus ๐ฅ is negative three ๐ฅ plus 19.

In this video, weโve seen that a
linear function is an algebraic equation which gives the graph of a straight
line. Weโve seen that they consist of an
input, thatโs usually ๐ฅ, and an output, thatโs usually ๐ฆ or ๐ of ๐ฅ. And that this input will never be
associated with exponents or powers. We learned that each input to the
function, which we treat a little like a function machine, has exactly one
output. And we find the outputs to our
function by substituting ๐ฅ. We might look to replace ๐ฅ with a
constant or another expression.