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
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
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
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.