Video: Completing Function Tables

Video Transcript

In this video, we will learn how to complete a function table for a linear function. We will begin by defining what we mean by some of the key terms.

A function table displays the inputs and corresponding outputs of a function. If we consider the function machine shown, we need to add seven to the input to calculate the output. Creating a function table as shown allows us to substitute different values for the input and then calculate the corresponding output. If we select the inputs three, six, and 10, we can calculate their corresponding outputs by adding seven to each of the numbers. Three plus seven is equal to 10, six plus seven is equal to 13, and 10 plus seven is equal to 17. The inputs three, six, and 10 in this function give us outputs of 10, 13, and 17, respectively.

A linear function is a function whose graph is a straight line. It is usually written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept. A linear function has one independent variable 𝑥 and one dependent variable 𝑦. Our 𝑥-values are the inputs, and our corresponding 𝑦-values are the outputs. If we consider the function machine and table above, we can replace input with 𝑥 and the output with 𝑦. As our rule was adding seven, our linear function is 𝑦 equals 𝑥 plus seven.

We will now look at some questions where we need to complete function tables given their linear function or rule.

Complete the given function table.

We’re given in the table that our input is 𝑥 and our output is two plus 𝑥. If we let the output be the letter 𝑦, we can see that we have the linear function 𝑦 is equal to two plus 𝑥. We know this is linear as the highest power of 𝑥 is one. We can now calculate our output values by substituting the input for 𝑥. When 𝑥 is equal to nine, 𝑦 is equal to two plus nine. This is equal to 11. So when the input is nine, the output is 11.

We can repeat this process when 𝑥 is equal to five. Two plus five is equal to seven. This means that when the input is five, the output is seven. Finally, we have an input of 16. 𝑦 is therefore equal to two plus 16, which is 18. When our input is 16, our output is 18. The linear function 𝑦 equals two plus 𝑥 means that each value of 𝑦 or the output is two greater than the value of 𝑥 or the input. Our correct answers are 11, seven, and 18.

Our next question involves a more complicated linear function.

Use the rule to find the missing number in this input–output table. Rule equals three 𝑛 minus seven.

We are told that when the input is four, the output is five. To check this, we can substitute the value 𝑛 equals four into the equation. Three 𝑛 means three times 𝑛, so we are left with three multiplied by four minus seven. Three multiplied by four is 12. And taking away seven does give us five. Our second set of values are an input of five and an output of eight. Substituting five into the rule does give us eight, as three multiplied by five is 15 and subtracting seven from this is eight. We can do the same for an input of seven, giving us an output of 14.

We need to calculate the missing value in the table. This is the output when the input is six. We can calculate the output by substituting 𝑛 equals six into our rule. Three multiplied by six is equal to 18. Subtracting seven from this gives us an answer of 11. When our input is six, our output is 11.

An alternative method to solve this question would be to notice that our rule, three 𝑛 minus seven, is a linear function. This means that for every increase of one to the input, the output will increase or decrease by the same value. This value is the coefficient of 𝑛, the number we’re multiplying by 𝑛, as this would be the slope or gradient of the linear function. We can see this in our function table as five plus three is equal to eight, eight plus three equals 11, and 11 plus three is 14. As the input increases by one, the corresponding output increases by three.

In our next question, we need to identify which table corresponds to a linear function.

Which of the following is a table of values for the function 𝑦 equals seven 𝑥 minus three?

There are a couple of ways of approaching this question. The quickest way would be to recognize that we have a linear function of the form 𝑦 equals 𝑚𝑥 plus 𝑏. We know that a linear function is a straight-line graph, and the value of 𝑚 is the gradient or slope of the line. This is the increase in 𝑦-values for each increase of one in the 𝑥-values. In this question, our slope or gradient is seven. The value of 𝑏 is the 𝑦-intercept of the graph. This is the point at which the graph crosses the 𝑦-axis. When 𝑥 is equal to zero, this is the 𝑦-value. In the equation 𝑦 equals seven 𝑥 minus three, the 𝑦-intercept is negative three. This means that our table must contain the corresponding values 𝑥 equals zero and 𝑦 equals negative three.

Options (A), (B), (C), and (E) all contain this value. Option (D), on the other hand, contains the value zero, three. When 𝑥 equals zero, 𝑦 is equal to three. Therefore, this is not the correct table. As the 𝑥-value in each of the tables increases by one, the 𝑦-value needs to increase by seven. In table (A), negative three plus seven is equal to four, four plus seven is equal to 11, and 11 plus seven is equal to 18. This means that table (A) does satisfy the function 𝑦 equals seven 𝑥 minus three. Option (B) is not correct as the difference between four and 10 is six and between 10 and 18 is eight. This means that table (B) does not represent a linear function and is therefore not equal to 𝑦 equals seven 𝑥 minus three.

We have a similar problem with table (C). Negative three plus six is equal to three. And three plus eight is equal to 11. Therefore, this function is not linear. In option (E), our first two steps are correct. Negative three plus seven is equal to four, and four plus seven is equal to 11. However, the final number, 19, should be 18 as 11 plus eight is equal to 19. The correct answer is option (A). The function 𝑦 equals seven 𝑥 minus three has coordinates zero, negative three; one, four; two, 11; and three, 18 that lie on the straight line.

An alternative method here would be to substitute our 𝑥-values, zero, one, two, and three, into the function 𝑦 equals seven 𝑥 minus three. When 𝑥 is equal to zero, 𝑦 is equal to seven multiplied by zero minus three. This is equal to negative three. Therefore, when 𝑥 is zero, 𝑦 is negative three. When 𝑥 is equal to one, 𝑦 is equal to four as seven multiplied by one minus three is four. Substituting in 𝑥 equals two gives us a value of 𝑦 equal to 11. Finally, when 𝑥 is equal to three, 𝑦 is equal to 18. This confirms that the four values of 𝑥 and 𝑦 need to be zero, negative three; one, four; two, 11; and three, 18.

Our next two questions require us to find missing 𝑥- and 𝑦-values in a function table.

Given that 𝑦 is equal to 40 divided by 𝑥, complete the following table.

There’re three missing values in the table. We need to calculate 𝑦 when 𝑥 is eight, 𝑥 when 𝑦 is two, and 𝑦 when 𝑥 is 40. The first and third missing numbers are relatively straightforward. Substituting 𝑥 equals eight into our equation or function gives us 𝑦 is equal to 40 divided by eight. This is equal to five. So our first missing value is five. We can repeat this process for the third missing value. Substituting 𝑥 equals 40 gives us 𝑦 is equal to 40 divided by 40. Dividing any number by itself gives us an answer of one. When 𝑥 is equal to 40, 𝑦 is equal to one.

For the second missing value in our table, we are given the value of 𝑦 and need to calculate the value of 𝑥. This means that two is equal to 40 divided by 𝑥. We can rewrite this as 40 over 𝑥 and then solve the equation to calculate 𝑥. Multiplying both sides of the equation by 𝑥 gives us two 𝑥 is equal to 40. Dividing both sides of this equation by two gives us 𝑥 is equal to 20. When 𝑦 is equal to two, 𝑥 is equal to 20. We can check this answer by dividing 40 by 20. As this is equal to two, our answer is correct. The missing numbers are five, 20, and one.

Given that 𝑦 is equal to two 𝑥 minus five, complete the following table.

We need to calculate the 𝑥-values that correspond to the 𝑦-values of 11, 19, and 30 in the linear function 𝑦 equals two 𝑥 minus five. The easiest way to do this is to substitute each of the 𝑦-values in turn into the function. We will begin with 𝑦 equals 11. This gives us 11 is equal to two 𝑥 minus five. We can solve this equation using the balancing method, firstly, by adding five to both sides. 11 plus five is equal to 16, so we have 16 is equal to two 𝑥. Our final step is to divide both sides of this equation by two. This gives us a value of 𝑥 equal to eight. When 𝑦 is equal to 11, 𝑥 is equal to eight.

We can repeat this process when 𝑦 is equal to 19. Adding five to both sides gives us 24 is equal to two 𝑥. Dividing by two gives us a value of 𝑥 equal to 12. When 𝑦 is equal to 19, 𝑥 equals 12. The same method can then be applied for our final missing value when 𝑦 is equal to 30. Once again, we follow a two-step process. We add five and then divide by two.

Adding five to both sides of the equation gives us 35 is equal to two 𝑥. Two does not divide exactly into 35 as 35 is an odd number. We could leave this answer as a top-heavy or improper fraction or change it to the decimal 17.5 as this is half of 35. When 𝑦 is equal to 30, 𝑥 is equal to 17.5. The three missing values in the table are eight, 12, and 17.5. We could substitute each of these values back into the function 𝑦 equals two 𝑥 minus five and ensure we get answers of 11, 19, and 30.

An alternative method here would be to consider function machines. Our value of 𝑥 is our input and our value of 𝑦 is our output. In the function 𝑦 equals two 𝑥 minus five, there are two steps. We multiply 𝑥 by two and then subtract five. As we need to calculate the 𝑥-values, we need to reverse these functions. The inverse or opposite of subtracting five is adding five, and the inverse of multiplying by two is dividing by two. In order to calculate our 𝑥-values, we need to add five to our 𝑦-value and then divide the answer by two. This is the same two steps that we completed when solving the equation. We added five to 11 and then divided by two, giving us an answer of eight.

We will now summarize the key points from this video. A function is a rule that assigns a set of inputs to a set of outputs. We looked at linear functions which are written in the form 𝑦 equals 𝑚𝑥 plus 𝑏. These are straight-line graphs where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept. Our 𝑥-values correspond to the inputs, and our 𝑦-values correspond to the outputs. We completed a variety of function tables by using rules and also by substitution and then solving equations. When we were given the output and needed to calculate the input, we used inverse functions. This involved using our knowledge of inverse or opposite operations.