The function notation represents the algebraic rule that applies to a given function. Through function notation, it is much easier to represent and model the rules that define a given set of input and output values. This topic is an essential building block to master when studying functions and algebraic models.
This article covers all the fundamentals needed to understand the significance of functions notations. In this discussion, you’ll also learn how to evaluate and model functions notations. Try out a wide range of examples to master this topic as well!
What is a Function Notation?
The function notation is a simple and efficient way to write functions making them easy to write, interpret, and evaluate. The function notation uses algebraic expressions to summarize how a function affects the input value to attain the desired output values. One of the most used function notations in math is \boldsymbol{f(x)},or read as “f of x”. The function, f(x) represents the function that defines the relationship between the input and output values.
Think of function and function notation like the products made in a factory machine’s production line. The input value, x, acts as the raw materials or products to be assembled into the machine. Once assigned to the proper assembly line, rules or steps are followed to meet certain output or end product. Now, think of f(x) as the output after the materials have gone through the process defined by the rules for f. There are instances when we use y to represent the output values, so use y =f(x) when necessary.
Components of Function Notation
Now, it’s time for us to understand the components of a function notation. Take a look at the notation of the function, f(x), which multiplies its input value by 5 then add the result by 1. It will be tedious to repeatedly define the rules for this function and this is why function notations like the one shown below are important.
The expression, f(x), shows that the function is defined by the rules of f (or any other letter) and with x as still the input of the function. The equation on the right-hand side highlights the process performed on the input value. For each value of x, the function, f(x) is will return output values that are defined by the rule, 5x + 1. Here are other examples of functions written in its established function notation:
- f(x) = 4x^2 – 4x + 4
- g(x) = x^3 - 1
- h(x) = 2x^3 – 4x^2 + 1
Since function notations are simply conventions or standards we use, there are in fact other ways to write functions. Here are some other function notations that you may encounter:
| f(x) = 4x^2 – 4x + 4 | g(x) = x^3 -1 | h(x) = 2x^3 – 4x^2 + 1 |
| y = 4x^2 – 4x + 4 f(x) =\{(x, y)|y = 4x^2 – 4x + 4\} x\mapsto 4x^2 – 4x + 4 | y = x^3 -1 f(x) =\{(x, y)|y = x^3 -1\} x\mapsto x^3 -1 | y = 2x^3 – 4x^2 + 1 f(x) =\{(x, y)|y = 2x^3 – 4x^2 + 1 \} x\mapsto 2x^3 – 4x^2 + 1 |
For the second notation, f(x) =\{(x, y)|y = 4x^2 – 4x + 4\}, the symbol “|” is read as “such that”. This means that the function is defined by x and y such that they follow the given rules for the function. Meanwhile, the third notation shown has the unique arrow symbol, “↦”. This simply shows that the input value is mapped or matched to the given function’s rules. For now though, our discussion will focus on the most standard function notation.
Problem 1
Apply what you’ve learned to write the function notation for the following statements describing each function.
Given the input value, x, the output value of the function, f, is the result of multiplying
the input value by -6 then dividing the result by 5.
Since f defines the function’s rule, express f(x) as an algebraic expression in terms of x. After multiplying x by -6, the resulting expression becomes -6x. Dividing this expression by 5 leads to the expression, \dfrac{-6x}{5} .
This means that the function being described has the following notation: f(X) = -\dfrac{6x}{5}.
Problem 2
Observe the table of values shown below then find the function notation for g(x).
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| g(x) | 18 | 8 | 2 | 0 | 2 | 8 | 18 |
First, find a rule that turns the input values (first row) into their corresponding output values (second row).
\begin{aligned}18 = 2 \times(-3)^2&,18 = 2 \times(3)^2\\8= 2 \times(-2)^2&,18 = 2 \times(2)^2\\2 = 2 \times(-1)^2&,2 = 2 \times(1)^2\end{aligned}This means that the function is defined by twice the square of the input value. To generalize this rule and write the function notation for g(x), we have g(x) = 2\cdot x^2 or g(x) = 2x^2. Now that the we’ve established what function notations are, it’s time to learn how to evaluate different functions.
How To Evaluate Function Notation’s Values?
When evaluating function notations for a particular input value, simply substitute the input value into the algebraic expression that defines the function. Remember the assembly machine in the previous section? When evaluating function notation’s value, there is a focus on one particular input and its corresponding output value.
Take a look at the function shown above – when evaluating f(X) = 3x + 1 at x = 4, substitute the given input value to evaluate and find f(4)’s value.
\begin{aligned}f(4) &= 3(4) + 1\\&= 12 + 1\\&= 13\end{aligned}This confirms that when evaluated at x = 4, f(x) = 3x + 1 is equal to 13. Now, apply a similar process to evaluate the following functions.
Problem 3
Evaluate the following functions at the specific input value given for each item.
a. f(x) = -4x^2, x= -2
Just like what we’ve shown in our discussion when evaluating functions at a specific input value, simply substitute the value of x into the expression for f(x).
\begin{aligned}f(-2) &= -4(-2)^2\\&= -4(4) \\&= -16\end{aligned}Apply the appropriate rules of operation when evaluating function notations. For this problem, we have f(-2) = -16.
b. g(x) = 2x^2 – 4x + 1, x= 0
Similar with the previous item, simply substitute x =0 into the expression that defines g(x).
\begin{aligned}g(0) &= 2(0)^2 – 4(0) + 1\\&= 2(0) – 0 + 1 \\&=0 -0 +1\\&=1 \end{aligned}When given zero as an input value, all the algebraic terms containing x becomes zero. Remember this because it simplifies the process of evaluating functions at x =0. Hence, g(0) = 1.
c. h(x) = \dfrac{1}{x - 1}, x= -4
As long as the input value is within the allowed values for the function, apply the same process. This means that to find h(-4)’s value, substitute x= -4 into the expression.
\begin{aligned}h(-4) &= \dfrac{1}{-4 – 1}\\&= \dfrac{1}{-5}\\&= -\dfrac{1}{5} \end{aligned}This shows that h(-4) is equal to -\dfrac{1}{5}.
What are the Applications of Function Notation?
The function notation has a wide range of applications including modeling relationships between two or more values and solving word problems involving functions. Here are some common applications of function notation:
- Finding significant values of a given function.
- Defining the relationship between two set of values – with one set as input values and the other set as output values.
- Observing given patterns and using their function notation to find key quantities.
Seeing how varied the applications of function notation are, it’s important to practice how to represent models as functions and how to evaluate functions at important input values.
Problem 4
Harold runs a co-working space where he rents out cubicle spaces to freelancers at a given hourly rate. Suppose that the total cost of using a cubicle for h hours is defined by the function, C in dollars as shown below.
\begin{aligned}C(h) = 2.50 + 3.50h\end{aligned}a. What does 2.50 represent?
First, observe what happens when the function is evaluated at x =0.
\begin{aligned}C(0) &= 2.50 + 3.50(0)\\&= 2.50 + 0\\&= 3.50\end{aligned}This means that when h =0, C(0) = 2.50. Regardless of the number of hours, the value, 2.50, will remain constant. When you rent the space, there is an initial charge of 2.50 dollars. This is what 2.50 represents – the initial charge.
b. Interpret the evaluated function notation, C(10) = 37.50, in this context.
For C(10) = 37.50, the input value is equal to 10 while the evaluated value of the function is 37.50. The input value represents the total number of hours spent at Harold’s co-working space and 37.50 represents the total cost.
This means that it costs 37.50 dollars to stay at the co-working space for ten hours.
c. If Celine books a cubicle the whole work week (Monday to Friday) and spends at co-working space 6 hours each day, how much will it cost her?
To find Celine’s total bill, find the total number of hours she spends in a work week at the co-working space. Since she spends 6 hours per day from Monday to Friday, Celine needs to book a total of 30 hours. Use this as the input value and evaluate the function notation that represents the cost.
\begin{aligned}C(30) &= 2.50 + 3.50(30)\\&= 2.50 + 105\\&= 107.50\end{aligned}This means that it will cost Celine a total of 107.50 dollars.
Problem 5
The function, g(x), has the graph shown below. The input values are represented by x while the output values are represented by y in the Cartesian plane shown below.
a. What is the function notation defining the function shown above?
To find the expression for the function, g(x), observe a pattern followed by the corresponding points. The table of values below summarizes the input and output values of the function from its coordinates.
| x | 2 | 4 | 6 | 8 | 10 |
| g(x) | 16 | 4 | 0 | 4 | 16 |
Try to think of a rule for g(x) so that these five pairs of input and output values remain true. The graph shown above is a quadratic function, so expect g(x) to be a quadratic expression (meaning the function will have a squared term).
\begin{aligned}16 = &(-4)^2= (2 - 6)^2\\4 = &(-2)^2= (4 - 6)^2\\0 = &(0)^2= (2 - 6)^2\\4= &(2)^2= (8 - 6)^2\\16 = &(4)^2= (10 - 6)^2 \end{aligned}This means that the function notation that defines the curve is g(x) = (x – 6)^2.
b. What is the value of the function if the input value is equal to 20?
To find the function’s value when x = 10, substitute x with 10 then simplify the expression for g(20).
\begin{aligned}g(20) &=(20 – 6)^2\\&= 14^2\\&= 196\end{aligned}This means that when x = 20, g(20) = 196. When the graph is extended, we expect it to pass through the point, (20, 196).