4.4.2 Defining Functions by a Rule

Function Notation

For the function $y=f(x)$:

• $f$ is the name of the function.
• $x$ is the input value.
• $f(x)$ is the output value, $y$, corresponding to the value of $x$.

We read $f(x)$ as $f$ of $x$ or the value of $f$ at $x$.

We call $x$ the independent variable as it can be any value in the domain. We call $y$ the dependent variable as its value depends on $x$.

Independent and Dependent Variables

For the function $y=f(x)$,

$x$ is the independent variable as it can be any value.

$y$ is the dependent variable as its value depends on $x$.

Much as when you first encountered the variable $x$, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.

Let’s look at the equation $y=4x-5$. To find the value of $y$ when $x=2$, we know to substitute $x=2$ into the equation and then simplify.

The value of the function at $x=2$ is 3.

We can do the same thing using function notation. The equation $y=4x-5$ can be written as $f(x)=4x-5$. To find the value when $x=2$, we write:

The value of the function at $x=2$ is 3.

This process of finding the value of $f(x)$ for a given value of $x$ is called evaluating the function.