In this video, we will learn how to calculate the value or output of a function using its equation or graph. We will begin by recalling what we mean by a function. And then we will explain how we can evaluate them.
The most popular function notation is 𝑓 of 𝑥. The 𝑓 of 𝑥 notation is another way of representing the 𝑦-value in a function such that 𝑦 is equal to 𝑓 of 𝑥. Instead of writing 𝑦 equals three 𝑥 plus seven, we will write 𝑓 of 𝑥 is equal to three 𝑥 plus seven. The 𝑦-axis may be labeled as the 𝑓-of-𝑥 axis when graphing the function. Labeling a function in this way avoids confusion when we’re dealing with multiple functions.
We will now look at how we can evaluate a function. To evaluate a function, we substitute the input or given number for the function’s variable, usually 𝑥. For example, let’s consider the function 𝑓 of 𝑥 is equal to three 𝑥 plus seven. If we were asked to evaluate 𝑓 of four, then four is the input. We substitute this into the expression. In this case, we have three multiplied by four plus seven. This is equal to 19, which is known as the output. When the input to the function is four, the output is 19. This can be written as an ordered pair or coordinate four, 19, where four is the 𝑥-value and 19 is the 𝑓-of-𝑥 or 𝑦-value.
We will now look at some questions that involve evaluating functions.
Using the function 𝑦 equals 𝑥 squared plus three, calculate the corresponding output for an input of two.
The output of any function in this format is the 𝑦-value. The input is the 𝑥-value. We need to substitute 𝑥 is equal to two into the function. As 𝑦 is equal to 𝑥 squared plus three, when 𝑥 is two, 𝑦 is equal to two squared plus three. Two squared is equal to four. So 𝑦 is equal to four plus three. As this is equal to seven, the output that corresponds to an input of two is seven.
We could write this as an ordered pair with coordinates two, seven. The point with coordinates two, seven lies on the quadratic function 𝑦 equals 𝑥 squared plus three. When dealing with functions, the 𝑦 is often replaced with 𝑓 of 𝑥. These are interchangeable, and both correspond to the output.
Our next question involves completing a table of values for a function.
Complete the table of values for the function 𝑦 equals three 𝑥 squared minus two 𝑥.
In this question, we’re given a function 𝑦 equals three 𝑥 squared minus two 𝑥 and five integer values of 𝑥 from negative two to two. In order to complete the table, we need to substitute each of these values in turn into the function. Let’s begin with positive two. When our 𝑥-value or input is equal to two, then our 𝑦-value or output will be equal to three multiplied by two squared minus two multiplied by two. Using our order of operations, three multiplied by two squared is equal to 12. We need to square the two and then multiply by three. Two multiplied by two is four. So we have 12 minus four. This is equal to eight. When 𝑥 is equal to two, 𝑦 is equal to eight.
We can repeat this process when 𝑥 is equal to one. Three multiplied by one squared is three, and two multiplied by one is two. As three minus two is equal to one, when 𝑥 is equal to one, 𝑦 is equal to one.
When 𝑥 is equal to zero, 𝑦 is equal to three multiplied by zero squared minus two multiplied by zero. Both parts of this calculation are equal to zero. And zero minus zero is zero.
We now need to consider when 𝑥 is negative, which is slightly more complicated. Squaring a negative number gives a positive answer, as multiplying a negative by a negative is a positive. This means that three multiplied by negative one squared is equal to three. Two multiplied by negative one is negative two. But as we’re subtracting this, we’re left with positive two. Three plus two is equal to five. So when 𝑥 is equal to negative one, 𝑦 is equal to five.
Three multiplied by negative two squared is 12, as negative two squared is four. As two multiplied by negative two is negative four, we need to add four to 12. Once again, we’re subtracting a negative number. This gives us an output or 𝑦-value of 16 when 𝑥 is negative two.
The five missing values in the table are 16, five, zero, one, and eight. We could use these coordinate pairs negative two, 16; negative one, five; and so on to graph the function 𝑦 equals three 𝑥 squared minus two 𝑥. As our function is quadratic and the coefficient of 𝑥 squared is positive, we will have a U-shaped parabola.
In our next question, we will need to identify which point satisfies a given function.
Which of the following set of coordinates lies on 𝑓 of 𝑥 is equal to negative 19𝑥 minus 16? Is it (A) 10, negative 16; (B) 10, negative 206; (C) negative 206, 10; or (D) negative 206, negative 16?
We are used to any pair of coordinates being written in the form 𝑥, 𝑦. The first coordinate is the 𝑥-value, and the second is the 𝑦-value. The function 𝑓 of 𝑥 is equal to negative 19𝑥 minus 16 is the same as 𝑦 equals negative 19𝑥 minus 16. As 𝑦 and 𝑓 of 𝑥 are interchangeable, we can write the coordinate pair as 𝑥, 𝑓 of 𝑥. The first value is known as the input and the second value the output.
We need to work out which of the four options satisfies the function. We notice that both option (A) and option (B) have an input or 𝑥-value of 10. This means that in order to calculate the output, we need to work out 𝑓 of 10. Substituting in 𝑥 equals 10 gives us negative 19 multiplied by 10 minus 16. Multiplying a negative number by a positive gives a negative answer. So negative 19 multiplied by 10 is negative 190. Subtracting 16 from this gives us negative 206. This means that the coordinate pair 10, negative 206 lies on the function. As option (A) was 10, negative 16, this is incorrect. Option (B), on the other hand, was 10, negative 206. So this is the correct answer.
We could check to see if options (C) and (D) are correct by substituting negative 206 in for 𝑥. 𝑓 of negative 206 is equal to negative 19 multiplied by negative 206 minus 16. We can clearly see that this will not satisfy either of our options, as our value is too large. An input of negative 206 gives an output of 3898. The coordinate pair negative 206, 3898 lies on the function. This does not correspond to option (C) or (D). So these are both incorrect. The correct answer is option (B).
Our next example is a multipart question involving the meaning of a function.
Given the function 𝑓, the meaning of 𝑓 of 𝑎 minus one is the output when the input is one less than 𝑎. Interpret the following. 𝑓 of 𝑏 plus three, 𝑓 of 𝑠 minus three, 𝑓 of three minus 𝑥, 𝑓 of 𝑏 minus 𝑓 of 𝑎, 𝑓 of three 𝑡, and 𝑓 of 𝑎 to the power 𝑏.
Before starting this question, it is worth recalling what we mean by a function 𝑓. If we have any function 𝑓 of 𝑥, then 𝑥 is the input and 𝑓 of 𝑥 is the output. A number inside the bracket affects the input, whereas a number outside of the bracket affects the output. This can be seen from the example, as 𝑎 minus one means one less than 𝑎.
Our first function, 𝑓 of 𝑏 plus three, is very similar to the example. Instead of subtracting one from 𝑎, we’re adding three to 𝑏. This means that 𝑓 of 𝑏 plus three calculates the output when the input is three more than 𝑏.
Our second function, 𝑓 of 𝑠 minus three, is slightly different. This time, the three that is being subtracted is outside of the bracket. 𝑓 of 𝑠 will be the output when the input is 𝑠. Therefore, 𝑓 of 𝑠 minus three is three less than the output when the input is 𝑠.
Our third function, 𝑓 of three minus 𝑥, is very similar to the first one and also the example. This time, our function gives us the output when the input is 𝑥 less than three. We are subtracting 𝑥 from three and then working out the output.
Our fourth function has two variables, 𝑏 and 𝑎. We have 𝑓 of 𝑏 minus 𝑓 of 𝑎. This means that we are subtracting the value of 𝑓 of 𝑎 from the value of 𝑓 of 𝑏. This is the difference between them. Therefore, the answer corresponds to the change in output when the input changes from 𝑎 to 𝑏.
The penultimate function is 𝑓 of three 𝑡. We are multiplying our value of 𝑡 by three. This is a similar idea once again to our first function, 𝑓 of 𝑏 plus three, and also our third function, 𝑓 of three minus 𝑥. This time, it corresponds to the output when the input is three times 𝑡.
Our final function involves exponents or powers. We have 𝑓 of 𝑎 to the power of 𝑏. As the 𝑏 is outside of the bracket or parentheses, this is the result of raising the output at input 𝑎 to the 𝑏th power. If the power 𝑏 was inside the bracket, we would be raising the input to the 𝑏th power. Interpreting and often drawing functions of this type is an important part of the topic.
Our final question involves evaluating a function from a graph.
Determine 𝑓 of one.
We can see from the graph that our axes are labeled 𝑥 and 𝑓 of 𝑥. Any function written in the form 𝑓 of 𝑥 has input 𝑥 and output 𝑓 of 𝑥. In this question, our value of 𝑥 or our input is one. We need to find the value of 𝑓 of 𝑥 from the graph when 𝑥 is equal to one. We do this by drawing a vertical line from one on the 𝑥-axis. Once we reach our graph, we draw a horizontal line across to the 𝑦- or 𝑓-of-𝑥 axis. This is equal to eight. Therefore, the value of 𝑓 of one is eight. We could work out the value of 𝑓 of negative two up to 𝑓 of eight using this graph.
We will now finish this video by summarizing the key points. As mentioned at the start of the video, the 𝑓 of 𝑥 notation is another way of representing the 𝑦-value of a function such that 𝑦 is equal to 𝑓 of 𝑥. If we consider the function 𝑓 of 𝑥 equals five 𝑥 minus two, then our 𝑥-value is the input. The value of five 𝑥 minus two is the output value. To evaluate a function, we substitute the input for the function’s variable. For example, to calculate 𝑓 of three, we substitute three for 𝑥. Five multiplied by three minus two is 13. Therefore, the input of three gives an output of 13.
We have seen from this video that we can calculate an output using equations, sometimes in the form of a table, or alternatively from graphs.