Lesson Video: Linear Functions Mathematics • Third Year of Preparatory School

Video Transcript

In this video, we’ll 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 𝑥. 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 𝑓 of 𝑥. 𝑓 is the name of the function, and the 𝑥 is what goes into it; it’s its input. In this function, for example, we put 𝑥 in, multiply it by two, and then subtract one. We could replace the 𝑥 with a number, such as five, and we’d 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. That’s 𝑓 of 𝑥 is equal to 𝑎𝑥 plus 𝑏 for real constants 𝑎 and 𝑏. Letting 𝑦 be equal to 𝑓 of 𝑥, these might look like this: 𝑦 plus 𝑥 equals five or three 𝑥 minus 𝑦 plus seven equals zero. These equations still correspond to the graph of a straight line; they’re still linear functions.

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

So, when 𝑥 is equal to nine, we replace — remember, the mathematical word for this is “substitute” — 𝑥 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, we’re going to input 𝑥 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 𝑥 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 𝑥. They’re equal to 11, 7, and 18. Now, this is, in fact, an example of a linear function. Let’s define our output as being equal to 𝑦. So, 𝑦 is equal to two plus 𝑥. Then, our pairs of values correspond to coordinates. When 𝑥 is equal to nine, 𝑦 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, we’ll look at an alternative way that we could present a function.

Find the value of 𝑓 of eight given the function 𝑓 of 𝑥 equals three minus seven 𝑥.

In this question, we’ve been given a function whose name is 𝑓. The 𝑥 is what goes into the function; it’s its input. We’re looking to find the value of 𝑓 of eight. Remember, the bit inside the parentheses or brackets is the input. So, what we’re really trying to find is what’s the output when the input is eight. So, we look at our function 𝑓 of 𝑥 equals three minus seven 𝑥. And each time we see an 𝑥, we replace or we substitute it with eight.

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

PEMDAS or BIDMAS, that’s 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. 𝑓 of eight is, therefore, three minus 56. Now, don’t change the order of the sum. We’re not calculating 56 minus three. And three minus 56 is negative 53. So, given the function 𝑓 of 𝑥 equals three minus seven 𝑥, the value of 𝑓 of eight is negative 53.

We’ll now look at an input–output table for a similar type of function.

Fill in the input–output table for the function 𝑦 equals five 𝑥 plus three.

Our table gives us a row of inputs. They’re zero, two, four, and five. Our job is to calculate the outputs given a specific function. The function we have is 𝑦 equals five 𝑥 plus three. The input is 𝑥, and 𝑦 is the output. There might be times you’ll see the output denoted as 𝑓 of 𝑥. This is essentially the same thing.

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

They are 𝑥 equals zero, 𝑥 equals two, 𝑥 equals four, and 𝑥 equals five. The function is 𝑦 equals five 𝑥 plus three. So, we begin by substituting 𝑥 equals zero. So, our function becomes 𝑦 equals five times zero plus three. That’s three. And so, the first output in our table is three.

Let’s repeat this process for our second input. This time, we’re going to replace 𝑥 with two. Our function becomes 𝑦 equals five times two plus three, which is equal to 13. And so, the second output is 13.

Our third input is 𝑥 equals four. And so, we get 𝑦 equals five times four plus three, which is equal to 23. We’ll 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, let’s 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 we’re likely working with a linear function. Remember, a linear function produces the graph of a straight line, so this makes a lot of sense.

We’ll now look at how to find an unknown given a linear function and an output.

Find the value of 𝑘 given 𝑓 of 𝑥 equals 𝑘𝑥 plus 13 and 𝑓 of eight is equal to negative 11.

Here is our function. 𝑓 is sort of the name of the function. And 𝑥 is what goes in; it’s its input. We’ve also been given some information about the value of 𝑓 of eight. In other words, our input is no longer 𝑥. It’s eight. And when the input is eight, the output is negative 11. So, let’s see what happens when we put eight into our definition of the function here.

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

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

The eight is multiplying the 𝑘. And so, the inverse operation we apply next is to divide both sides of our equation by eight. That leaves 𝑘 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 𝑓 of 𝑥 is equal to 𝑘𝑥 plus 13 and 𝑓 of eight is equal to negative 11, we find 𝑘 is equal to negative three.

Now, we can check our answer by letting 𝑓 of 𝑥 now be equal to negative three 𝑥 plus 13. We’ve replaced 𝑘 with negative three. We’re going to check that the value of 𝑓 of eight is indeed negative 11. And so, we replace 𝑥 with eight. And we get negative three times eight plus 13. That’s negative 24 plus 13, which is negative 11 as required.

We’ll 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 𝑥. And the three inputs we’ve been given are 𝑥 equals one, 𝑥 equals four, and 𝑥 equals 10. So, let’s imagine we’re 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 𝑥 equals 10, we get an output of 18.

We’re trying to find a general rule. So, our job is to work out what we get when we input 𝑥. So, what is happening each time? We can actually observe that we’re 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 𝑥, we add eight. And we simply write that as 𝑥 plus eight. And that’s our output. It’s 𝑥 plus eight.

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

Evaluate 𝑓 of four minus 𝑥, given that 𝑓 of 𝑥 is equal to three 𝑥 plus seven.

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

But we haven’t been given a single number. We’re told to evaluate 𝑓 of four minus 𝑥. And so, instead of replacing 𝑥 with a number like two, we’re actually just going to replace 𝑥 with our input, with four minus 𝑥. And so, 𝑓 of four minus 𝑥 becomes three times four minus 𝑥 plus seven.

Let’s distribute the parentheses or expand the brackets. To do so, we’re going to multiply the three by the four and the three by the negative 𝑥. Three multiplied by four is 12. And three multiplied by 𝑥 is three 𝑥. So, three multiplied by negative 𝑥 is negative three 𝑥. 𝑓 of four minus 𝑥 is, therefore, 12 minus three 𝑥 plus seven. We then collect like terms. 12 plus seven is 19. And so, we see that 𝑓 of four minus 𝑥 is negative three 𝑥 plus 19.

In this video, we’ve seen that a linear function is an algebraic equation which gives the graph of a straight line. We’ve seen that they consist of an input, that’s usually 𝑥, and an output, that’s usually 𝑦 or 𝑓 of 𝑥. 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 𝑥. We might look to replace 𝑥 with a constant or another expression.