### 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.