The notion of function

Functions: Introduction to functions

The notion of function

Functions

A (real) function assigns to a real number a single real number.

If we say that $f$ is the function $x^2$ , we mean that the function rule of this function is $f(x)=x^2$ . The left hand side of this equation indicated the value of $f$ at $x$ ; the right hand side of this equality makes the assigned value explicit. If we substitute $x=2$ , the equation becomes $f(2)=4$ , which tells us that $f$ assigns the value $4$ to $2$ .

Instead of the function rule $f(x)=x^2$ we also write the formula $y=x^2$ . The real number $y$ that the function $f$ assigns to a number $x$ is denoted by $f(x)$ and is called the value of (the function) $f$ at $x$ (it is pronounced as: $f$ of $x$ ) or the function value at $x$ . This value is also called the image of $x$ under $f$ .

The set of all real numbers that can be entered in the function is called the domain of the function. The domain of the function $g$ with function rule $g(x) = \frac{1}{x}$ cannot contain the number $0$ , since it is not possible to divide by $0$ . We then say that $g$ is not defined at $0$ .

We call $x$ (the argument) the independent variable and $y$ the dependent variable.

Other variables can occur in the function rule. For instance: $f(x) = a\cdot x^2+t$ . Here, $a$ and $t$ play the role of constants; these variables are called parameters.

The function rule is the expression telling us how to calculate the value $f(x)$ for each number $x$ of the domain. For instance, $f(2)$ is the value of $f$ at $2$ , but not the function rule.

The choice for these specific names of the variables $x$ and $y$ is a habit from which we can deviate whenever we wish. To avoid any confusion it is important to mention this explicitly. For instance, by not simply writing an expression like $a\cdot x^2+t$ , but by specifying $a\cdot x^2+t$ as a function of $a$ (or of $x$ , or of $t$ ). We then prefer to write $f(a)=a\cdot x^2+t$ (or $f(x)=a\cdot x^2+t$ , or $f(t)=a\cdot x^2+t$ , respectively). The other variables ( $x$ and $t$ in this example) are called parameters in order to distinguish between the argument $a$ of the function $f$ and the other variables appearing in the function rule. So the parameters play the role of constants.

Often, the function rule $x^2+x+1$ is also called a function. We then mean the function defined by the function rule.

If $f$ is a real function given by means of a function rule, then we often speak of the domain of $f$ when we mean the greatest possible domain in $\mathbb{R}$ for $f$ , that is, the set of all real numbers $x$ at which $f(x)$ is defined.

Consider the real function $f$ which assigns the element $5\cdot x^2$ to $x$ . Thus, the function is described by $f(x)=5\cdot x^2$ .

What is the value of $f$ at $-1$ ?

$f(-1) = 5$

The function value at $-1$ is $5\cdot \left(-1\right)^2$ , or $5$ .

At $1$ the value of $f$ is also equal to $5$ .

New example