Representations of Functions
In mathematics, a functional relationship depicts the relationship between two variables – dependent and independent or between a set of input and output values. The output value is determined for different values of the input and this means that the input variable is independent and the output variable is dependent. The relationship between both the values obtained for such related variables is referred to as a ‘function’.
Function Rule
The relationship between the set of inputs (or independent variable values) and the set of outputs (or dependent variable values) can be represented in an equation form. This equation is known as the function rule.
- The cause or input is the independent variable. Its value is unaffected by the function’s other variables. An independent variable is usually denoted by ‘x’.
- The effect or output is the dependent variable. Its value is determined by the changes in the independent variable. The dependent variable is usually denoted by ‘y’.
- A function rule is the equation which represents the relationship between the 2 variables (x and y) by expressing the dependent variable ‘y’ in terms of the independent variable ‘x’.
For instance; if mangoes are sold for 6 dollars per kilogram, then the total cost of mangoes sold will depend upon the quantity of mangoes sold. So the total cost is the dependent variable (y) and the quantity of mangoes is the independent variable (x).
The relation between the two variables will be given by the equation:
y = 6x
Hence, the function rule is y = 6x.
How to write the Function Rule?
A function rule can be expressed in words (verbal form), an equation, table or a graph.
For example,
1. For the rule, “the output is 2 times lesser than the input value”, the equation formed will be y = x – 2.
2. Similarly, for the rule in words, “the output is the cube of the input value”, the equation formed will be \( y=x^3 \).
How to Evaluate a Function?
Determining the value of the function f(x) or output y that corresponds to a specified value of x is known as evaluating a function. To evaluate a function, simply substitute the given value for the independent variable x and solve for the dependent variable y.
For example, evaluate y = 3x + 2 when x = 2.
y = 3x + 2 (writing the equation)
y = 3(2) + 2 (Substituting 2 for ‘x’)
y = 6 + 2
y = 8 (simplified)
How to represent Functions using Graphs and Tables?
A function can be represented as a graph and a table.
If a function is represented as a graph on a coordinate plane, the set of values of the two variables of the function are the coordinates of the points lying on that graph.
From the graph of a function we can read the output (or y) value for a given input value by reading the y-coordinate of the point corresponding to that input value.
- Assume we have a function y = -2x + 3 with y being the dependent variable and x as the independent variable. Take x = -2 and get the corresponding value of y. When x is -2, y is 7 (by simplifying the equation, -2(-2) + 3 = 7 ). Instead, if we had a graph for this function, the input-output pairs would be the coordinates of the points lying on the graph.
- Look at the graph of the equation y = -2x + 3,
- Read the y-coordinate of the point that corresponds to x = -2. The point (-2, 7) on the graph of this function indicates that the output is 7 when the input is -2.
- The input-output pairs can be shown directly in a table. This is the tabular representation of this function. Here again we can see when the input is -2, the output is 7.
x | 2 | 1 | 0 | -1 | -2 |
|---|---|---|---|---|---|
y | -1 | 1 | 3 | 5 | 7 |
Solved Examples
Example 1: Write an equation for the following function rule: ‘The output is twelve times greater than the input value’.
Solution:
Writing in words: The output is twelve times greater than the input value.
Writing in equation form, y = x + 12.
Example 2: Evaluate the function y = 5x + 4 when the value of x = 2.
Solution:
This means we evaluate the value of the function when the value of ‘x’ is set to 2.
y = 5x + 4 (write the equation)
y = 5(2) + 4 (Substituting 2 in the place of ‘x’)
y = 10 + 4
y = 14 (simplified)
Hence the value of function y is 14.
Example 3: Graph the function y = 4x + 1.
Solution:
To plot the graph, make an input-output table to get the points that will lie on the graph of the given function.
For this, take the value of x as -1, 0, 1. Substitute these values in the equation and determine the corresponding values of y.
We will get the table as :
x | -1 | 0 | 1 |
|---|---|---|---|
y | -3 | 1 | 5 |
Plot the points and draw a line through them. The resulting line is the graph of the given function.
Example 4: Mary visits an amusement park and pays $10 for admission and $5 for each ride. Write a function rule to represent the total amount paid by Mary?
Solution:
The total amount is calculated by multiplying the number of rides by $5 and adding $10 to it.
Hence the total cost is dependent on the number of rides. So this can be written as a function in equation form as:
y = 5x + 10 (where y is the total amount and x is number of rides).
Example 5: There is $300 in a bank account and every week an additional $30 is added to it. Write a function to represent the balance in the bank account every week. Also find the total amount in the account after six weeks.
Solution:
The balance is the sum of the initial amount ($300) and amount added over the next few weeks.
Hence, the total balance is dependent on the number of weeks. So this can be written as a function in equation form as:
m = 300 + 30w (where m is the total balance and w is the number of weeks)
After 6 weeks the balance amount will be
m = 300 + 30(6) (substitute w as 6)
m = 300 + 180 (multiply)
m = 480 (add)
Hence, the function can be written as m = 30w + 300 and the total balance amount in the bank after 6 weeks will be $480.
A mapping diagram is a way of representing a function rule. The set of inputs are paired with the set of outputs using two columns placed together. The values in the left column represent the input and the column on the right represents the output values.
The set of input or independent variable values in a function is called the domain of the function. Whereas the set of output or dependent variable values is called the range of that function.