What does an open dot mean in math?

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I am currently doing a math problem and have come across an unfamiliar notation. A mini circle between $f$ and $h(x)$

The question ask me to find for 'the functions $f(x)=2x-1$ and $h(x)=3x+2$'

$$f \circ h(x)$$

However, I can't do this as I do not know what the circle notation denotes to. Does it mean to multiply?

asked Dec 8, 2014 at 7:28

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This notation means that you take the output of $h$ and use it as the input of $f$. When we are working with a specific $x$ value, we can suggestively write $f(h(x))$ instead.

For instance if $f(z)=1/z$ and $h(x)=2+3x$ then $$(f\circ h)(x) = f\big(h(x)\big) = f(2+3x) = \frac{1}{2+3x}.$$

(Note: I only used $z$ as the variable for $f$ to avoid confusion; in practice the function does not care what its input variable is named.)

answered Dec 8, 2014 at 7:32

Eric StuckyEric Stucky

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The circle $\circ$ is the symbol for composition of functions. In General, if you have two functions $g\colon X\rightarrow Y$ and $f\colon Y\rightarrow Z$, then $f\circ g$ is a function from $X$ to $Z$. For $x\in X$ one has $(f\circ g)(x) = f(g(x))$.

In your case one has: $f(x) = 2x-1$, $g(x) = 3x+2$ and $$ (f\circ g)(x) = f(g(x)) = 2(g(x))-1 = 2(3x+2) -1 = 6x+3. $$ You take the function $g(x)$ and put it in place of the $x$ in the function $f$.

This is obviously different from $f(x)\cdot g(x) = (2x-1)\cdot (3x+2) = 6x^2+x-2$.

answered Dec 8, 2014 at 7:43

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"Function Composition" is applying one function to the results of another:

The result of f() is sent through g()

It is written: (g º f)(x)

Which means: g(f(x))

Example: f(x) = 2x+3 and g(x) = x2

"x" is just a placeholder. To avoid confusion let's just call it "input":

f(input) = 2(input)+3

g(input) = (input)2

Let's start:

(g º f)(x) = g(f(x))

First we apply f, then apply g to that result:

(g º f)(x) = (2x+3)2

What if we reverse the order of f and g?

(f º g)(x) = f(g(x))

First we apply g, then apply f to that result:

(f º g)(x) = 2x2+3

We get a different result!

When we reverse the order the result is rarely the same.

So be careful which function comes first.

Symbol

The symbol for composition is a small circle:

(g º f)(x)

It is not a filled in dot: (g · f)(x), as that means multiply.

Composed With Itself

We can even compose a function with itself!

Example: f(x) = 2x+3

(f º f)(x) = f(f(x))

First we apply f, then apply f to that result:

(f º f)(x) = 2(2x+3)+3 = 4x + 9

We should be able to do it without the pretty diagram:

(f º f)(x)= f(f(x))

 = f(2x+3)

 = 2(2x+3)+3

 = 4x + 9

Domains

It has been easy so far, but now we must consider the Domains of the functions.

The domain is the set of all the values that go into a function.

The function must work for all values we give it, so it is up to us to make sure we get the domain correct!

Example: the domain for √x (the square root of x)

We can't have the square root of a negative number (unless we use imaginary numbers, but we aren't), so we must exclude negative numbers:

The Domain of √x is all non-negative Real Numbers

On the Number Line it looks like:

Using set-builder notation it is written:

{ x

Or using interval notation it is:

[0,+∞)

It is important to get the Domain right, or we will get bad results!

Domain of Composite Function

We must get both Domains right (the composed function and the first function used).

When doing, for example, (g º f)(x) = g(f(x)):

• Make sure we get the Domain for f(x) right,
• Then also make sure that g(x) gets the correct Domain

Example: f(x) = √x and g(x) = x2

The Domain of f(x) = √x is all non-negative Real Numbers

The Domain of g(x) = x2 is all the Real Numbers

The composed function is:

(g º f)(x) = g(f(x))

 = (√x)2

 = x

Now, "x" normally has the Domain of all Real Numbers ...

... but because it is a composed function we must also consider f(x),

So the Domain is all non-negative Real Numbers

Why Both Domains?

Well, imagine the functions are machines ... the first one melts a hole with a flame (only for metal), the second one drills the hole a little bigger (works on wood or metal):

What we see at the end is a drilled hole, and we may think "that should work for wood or metal". But if we put wood into g º f then the first function f will make a fire and burn everything down!

So what happens "inside the machine" is important.

De-Composing Function

We can go the other way and break up a function into a composition of other functions.

Example: (x+1/x)2

That function can be made from these two functions:

f(x) = x + 1/x

g(x) = x2

And we get:

(g º f)(x) = g(f(x))

 = g(x + 1/x)

 = (x + 1/x)2

This can be useful if the original function is too complicated to work on.

Summary

• "Function Composition" is applying one function to the results of another.
• (g º f)(x) = g(f(x)), first apply f(), then apply g()
• We must also respect the domain of the first function
• Some functions can be de-composed into two (or more) simpler functions.

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