### 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 **v****alue 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 #5#?

#f(5) = 125#

The function value at #5# is #5\cdot \left(5\right)^2#, or #125#.

At #-5# the value of #f# is also equal to #125#.

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