什么是函数？

(Video) Introduction to Functions

by mathman1024

In this video, mathman1024 explains the basics of functions.

$\large{\text{Summary}}$

Here's an example of a relation: $$\{(2,3),(4,1),(0,1),(2,-2),(-1,5)\}$$

Sometimes when we put a bunch of ordered pairs together, we get a pattern that creates a graph.

Definition of a function

Now let's learn about functions. A function is a relation where each $x$ is paired with no more than one $y$. Note that the same $y$ can be paired with different $x$'s, but not the reverse. All linear equations, with the exception of vertical and horizontal lines, are functions.

Nope! This set of values can't correspond with a function because the $x$-value $2$ is paired with two different $y$ values: $3$ and $-2$.

If we look back at the graph of the function, we may be wondering what would happen if we found the inverse of the function, which could help later on.

Functions - What are Those?!

In math, think of a function like a little machine. We give it an input, like putting a penny into a gumball machine. The job of the function is to produce an output for each input we give it.

If the input is like a penny into a gumball machine, then the output is like the gumball. The function is the gumball machine itself—the thing changing the input to the output.

So, we obviously can't draw a gumball machine every time we want to write down a function in math. Instead, functions are written down like this: $f\left(x\right)$

No matter how we write our function, you can still think of it as out little gumball machine, giving an output for each input.

Let's look at an example of a simple function.

Note also that we don't have to use $x$ when we write our functions. You can use any letter you want.

The function above (in pink) says that "for each $x$ we put in, we get $2x$ out."

Let's try another, more complex example. $$f(x) = \frac{x - 1}{x + 2}$$

We can use functions for things other than real numbers, though. We can also have functions for complex numbers and square roots!

(Video) Functions and Relations

by Eddie Woo

Here's a video by Eddie Woo where he briefly goes over the fundamentals of functions.

$\large{\text{Summary}}$

What are Functions?

A function is a kind of rule that, for one input, it gives you one output.

An example of this would be $y=x^2$. If you put in anything for $x$, you get one output for $y$. We would say that $y$ is a function of $x$ since $x$ is the input value.

(Video) Function Notation

by Eddie Woo

Function Notation

The way to denote a function is by using the letter $f$. We then have parentheses that contain our input. Usually, we see an $x$ here. Then we set the function equal to something which is our output, or $y$. $$f(x)=y\\ \text{Name}(\text{Input})=\text{Output}$$

Note that these do not need to be $f,x,$ or $y$. For example: $$A(r)=\pi r^2$$ $A$ is what we're naming this function, because it stands for area. More specifically, area of a circle. This is a function because if we put in an input, there's only one output. We say $A$ is a function of $r$. Our input in this example is $r$, or the radius. The output is $\pi r^2$, or what the actual area is.

How function notation works is whatever number you see as the input, you substitute that into the equation for the respective variable. $$\begin{align} A(5)&=\pi({\color{#8925AE}{5}})^2\\ A(5)&=25\pi \end{align}$$

You can also have functions that have multiple inputs. $$V(r,h)=\pi r^2h$$ This is a function for the volume of a cylinder and its inputs are radius and height. We would say here that $V$ is a function of $r$ and $h$.

We can think of functions as a machine. We can plug in a value, and our machine will do something to it, and it will output a result. For example, $f(x)=2x+3$ is basically plugging in any number, which our machine will double and add $3$ to, yielding a result.

The $(x)$ that follows the $f$ means that we are putting $x$ into the function $f$. The $x$ is really a dummy variable, because it is just a placeholder. When we input a specific value, we replace $x$ with that value. For example, $f(5)$ using the function we set above would turn out to be $2(5)+3=13$ When we read $f(x)$ we say "f of x."

The domain of a function is all the values that can be put into the machine and result in an output. The range of a function is all the possible outputs.

Do you want to try an example?

Consider the function $f(x)=2x^2-5$.

What is the domain of this function? Try to figure this out yourself before continuing!

All real values will work for $x$, our function will always produce a result equal to $2x^2-5$. Therefore, our domain is $\mathbb{R}$!

Now, this is a bit trickier, but you will be able to do it $:)$

What is the range of $f$? Remember, the range is all the outputs of the function. Try to reason through it.

We know that $2x^2$ must be positive. Thus, $2x^2-5 \geq -5$. This means $f$ cannot output numbers smaller than $-5$. We can check that all the other values greater than or equal to $-5$ work, and when it all checks out, we know that the range is $[-5,\infty)$.

(Video) What are Functions?

by Math Antics

In this video, mathantics explains what a function is.

$\large{\text{Summary}}$

Function: Something that relates or connects one set to another set in a particular way.

Set: A group or collection of things. The notation for a set is $\{1, 2, 3...\}$, where the elements of a set are placed inside those brackets.

The two sets that are related in a function are called the input set and output set. The function takes each member of the input set and relates it or maps it to a value in the output set. The input set is usually called the domain and the output set is called the range.

Function notation

Functions are usually written as $f(x)=3x$ instead of $y=3x$. Here, the $f(x)$ does not mean $f\times x$. Instead, it just refers to the equation being a function. If we look at the function above, we would say, "$f$ of $x$ equals three $x$."

If a number is placed where the $x$ is, that number is your input. For example: $$\begin{align} f(x) &= 3x\\ f(4) &= 3x\\ f(4) &= 3({\color{Red}{4}})\\ f(4) &= 12 \end{align}$$