Function Definition Math
Function Definition Math
We now need to move into the second topic of this chapter. The first thing that
we need to do is define just what a function is. There are lots and lots of
definitions for a function out there and most of them involve the terms rule,
relation, or correspondence.
While these are more technically accurate than the definition that we're going to
use in this section all the fancy words used in the other definitions tend to just
confuse the issue and make it difficult to understand just what a function is.

So, here is the definition of function that we're going to use. Again, I need to point
out that this is NOT the most technically accurate definition of a function, but it is
a good "working definition" of a function that helps us to understand just how a
function works.

"Working Definition" of Function :- A function is an equation (this is where
most definitions use one of the words given above) for which any x that can be
plugged into the equation will yield exactly one y out of the equation.
Know More About :- Graphing Circles


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There it is. That is the definition of functions that we're going to use. Before we
examine this a little more note that we used the phrase "x that can be plugged
into" in the definition.
This tends to imply that not all x's can be plugged into an equation and this is in
fact correct. We will come back and discuss this in more detail towards the end of
this section, however at this point just remember that we can't divide by zero and
if we want real numbers out of the equation we can't take the square root of a
negative number. So, with these two examples it is clear that we will not always
be able to plug in every x into any equation.

When dealing with functions we are always going to assume that both x and y will
be real numbers. In other words, we are going to forget that we know anything
about complex numbers for a little bit while we deal with this section.

Okay, with that out of the way let's get back to the definition of a function. Now,
we started off by saying that we weren't going to make the definition confusing.
However, what we should have said was we'll try not to make it too confusing,
because no matter how we define it the definition is always going to be a little
confusing at first.

In order to clear up some of the confusion let's look at some examples of
equations that are functions and equations that aren't functions.
Example 1 Determine which of the following equations are functions and which
are not functions.
Solution :- The definition of function is saying is that if we take all possible values
of x and plug them into the equation and solve for y we will get exactly one value
for each value of x.
Learn More :- Graph A Line


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At this stage of the game it can be pretty difficult to actually show that an
equation is a function so we'll mostly talk our way through it. On the other hand
it's often quite easy to show that an equation isn't a function.

(a) So, we need to show that no matter what x we plug into the equation and
solve for y we will only get a single value of y. Note as well that the value of y will
probably be different for each value of x, although it doesn't have to be.

So, for each of these value of x we got a single value of y out of the equation.
Now, this isn't sufficient to claim that this is a function. In order to officially prove
that this is a function we need to show that this will work no matter which value of
x we plug into the equation.

Of course we can't plug all possible value of x into the equation. That just isn't
physically possible. However, let's go back and look at the ones that we did plug
in. For each x, upon plugging in, we first multiplied the x by 5 and then added 1
onto it. Now, if we multiply a number by 5 we will get a single value from the
multiplication.
Likewise, we will only get a single value if we add 1 onto a number. Therefore, it
seems plausible that based on the operations involved with plugging x into the
equation that we will only get a single value of y out of the equation.


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