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Home > Introduction to Pre-Calculus > Functions

Functions

A function is a relationship between two variables where for every independent variable, there is only one dependent variable. This means for every x value, there is only one y value. An independent variable is a variable which can take any value, whereas a dependent variable takes a value according to a set rule involving the independent variable.

For example, in the equation y = x + 1, x is the independent variable and y is the dependent variable

When x = 2, y = 2 + 1 = 3.

Such an equation involving a dependent (y) and independent (x) variable is called a function. It is expressed as y = f(x).

Here is an example of a relationship that is not a function.

 

non-function relationship

 

In this example, the ordered pairs are (A, 1), (B, 1), (C, 2), (C, 4) and (D, 3). Notice that C has two dependent values 2 and 4. This means it is not a function.

 

Another example of an equation that is not a function is that of a circle given by

x2 + y2 = r2, or

y2 = r2 – x2, or

y =  pm sqrt {r^2 - x^2}

For every value of x, there are 2 possible values of y,   + sqrt {r^2 - x^2} or  - sqrt {r^2 - x^2} .

Hence a circle is not a function.

 

Now let’s look at a few examples of functions