# Function Notation

Introduction to functions

Tip: This isn't the place to ask a question because the teacher can't reply.

1 of 2 videos by Khan Academy

## Key Questions

• Function Notation is a structure of symbols used in Mathematics.

It is a logical and constructive way of portraying different types of functions. Example: A linear function can be represented as $y = 2 x + 4$. In function notation, it is not written this way. it is rather written as $f \left(x\right) = 2 x + 4$. The only difference is that $y$ was changed to $f \left(x\right)$, which is usually said as '$f$ of $x$'.

To evaluate this, $x$ would be equal to 8 ($x = 8$). With function notation, this would be easily represented as find $f \left(8\right)$. This means place $x$ with 8 and solve: $f \left(8\right) = 2 \left(8\right) + 4 = 16 + 4 = 20$.

• An equation is an equality which is satisfied by a unique set of values of your variables. You have, after the $=$ sign a fixed value, a fixed result.

For example: the equation $4 x - 2 = 0$ has zero as result and only $x = \frac{1}{2}$ as solution; this means that if you substitute the value of $x = \frac{1}{2}$ in the equation you have the result zero, i.e., the equation is satisfied.

Now, a function is similar, the only difference is that now you can have a lot of results after the $=$ sign and so you can have a lot of solutions.

For example: the function $4 x - 2 = y$ doesn't have a definite result (as before that was zero) but another variable $y$, so every time you choose an $x$ you'll get the corresponding value of $y$ that satisfies it.
If you choose:
$x = 1 \to y = 2$
$x = 2 \to y = 6$
....etc.

If $x = \frac{1}{2} \to y = 0$ which is the solution that we found before for our specific equation (in which the $y$ was already set as zero)!

So to summarize, an equation has a fixed result (after the $=$ sign) and an unique set of solutions (values of the variables); a function can have a lot of results (possibly $\infty$) and, as a consequence, a lot of solutions.

hope it helps

• To differentiate between a function and a polynomial.
Eg:
$f \left(x\right) = 2 x + 3$ is a function.
$p \left(x\right) = 2 x + 3$ is a polynomial.

There's no hard rule for using letters for representing a function, but f and g are used the most for general functions. And for polynomials, any letter would do. In some cases, the letter written is a capital like this:

$P \left(x\right) = 2 x + 3$

This is called a mathematical statement. It is used in logical deductions and in probability distributions.

• The Independent Variable doesn't depend on another variable. In most cases it is time, and is usually expressed with the variable x. The Dependent Variable depends on the independent variable and is usually expressed with y.

An example would be Jimmy was running for 5 hours, at 1 hour he had traveled 5 miles, and at the 3rd hour he had ran 15 miles. What is the Independent and dependent variables?

The answer would be that the distance ran would be the dependent variable, and time would be the independent variable, because the time stays constant and the distant traveled relies on how much time has passed.

https://nces.ed.gov/nceskids/help/user_guide/graph/variables.asp

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